UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Transformer modelling for transient studies Neves, Washington L. A. 1994

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


831-ubc_1995-983329.pdf [ 2.93MB ]
JSON: 831-1.0065027.json
JSON-LD: 831-1.0065027-ld.json
RDF/XML (Pretty): 831-1.0065027-rdf.xml
RDF/JSON: 831-1.0065027-rdf.json
Turtle: 831-1.0065027-turtle.txt
N-Triples: 831-1.0065027-rdf-ntriples.txt
Original Record: 831-1.0065027-source.json
Full Text

Full Text

TRANSFORMER MODELLING FOR TRANSIENT STUDIESbyWASHINGTON L. A. NEVESB. Eng., Universidade Federal da Paraiba, 1979M.Sc., Universidade Federal da ParaIba, 1982A DISSERTATION SUBMITTED iN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYTNTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF ELECTRICAL ENGINEERINGWe accept this thesis a conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIANovember 1994© Washington L.A. Neves, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of E (eY,.iThe University of British ColumbiaVancouver, CanadaDateeetde ,gg4DE-6 (2/88)AbstractTransformer modelling is a concern for the utility industry. The object of this workis to develop and investigate dynamic core models suitable for transient studies. A majoradvantage of the core models developed here is that they use readily available transformertest data as supplied by the manufacturer.For ferroresonance and inrush current studies, core saturation needs to berepresented reasonably well. A direct approach to producing nonlinear peak flux-currentand voltage-current characteristic of the iron core, taking iron losses into account, ispresented. The algorithm is simple in concept, easy to implement, and may be useful forelectromagnetic transient programs. A crude estimation of the transformer open circuitcapacitance is also made from rated frequency data. It is useful for situations in which thetransformer exciting current experiences strong capacitive effects.An iterative algorithm for more correctly representing the flux-linkage curve of adelta-connected transformer, suitable for situations in which the tests are performed with aclosed delta, is developed. It uses positive sequence excitation test data as input and takesinto consideration the removal of triplen harmonics from the line current.An approach to model frequency-dependent effects in the transformer core fromtransformer no-load loss data, is presented. Hysteresis and eddy current effects in the coreare treated simultmeously. The flux-current trajectories are generated by circuit modelswith no need to pre-define them.Simulations using the developed models are compared to laboratory measurementof inrush current and to a ferroresonance field test.IITable ofContentsAbstractTable ofContentsList of Tables ViiiList ofFigures 1XAcknowledgement xiiiDedication XIV1 Introduction 11.1 Transformer Modelling 11.2 Thesis Outline 21.3 Thesis Contributions 32 Literature Review and Case Studies ‘I2.1 Introduction 42.2 EMTP Basic Models 42.3 Other Models 92.4 Iron Losses 121112.4.1 Laboratory Measurements 182.4.2 Building Factor 202.5 Transformer Core Representation During Transients 212.5.1 Ferroresonance Case Study 212.5.2 Inrush Current Case Study 242.6 Summary 253 On Modelling Iron Core Nonlinearities 283.1 Introduction 283.2 Saturation Curves 293.2.1 Computation of the v - i,. Curve 323.2.2 Computation of the A-i, Curve 363.3 Comparisons Between Experiments and Simulations 383.4 Ferroresonance Simulations and Field Test 413.5 Distribution Transformer Saturation Curves 433.6 Summary 454 Saturation Curves ofDelta-Connected Transformers FromMeasurements4.1 Introduction 474.2 Basic Considerations 484.3 Saturation Curves 49lv4.3.1 Computation of the v - 1T4 Curve 514.3.2 Computation of the 2 -44 Curve 534.4 Case Study 554.5 Summary 575 Hysteresis andEddy Current Losses in Iron Core5.1 Introduction 595.2 Frequency Dependent Core Models 595.3 Core Loss 625.4 Eddy Current and Hysteresis Modelling 635.4.1 Construction ofY(s) from Its Real Part 645.4.2 Linear Network Synthesis 665.4.3 Iron Core Nonlinearities 695.4.4 Numerical Example - Hysteresis 695.5 Inrush Current: Simulation and Measurements 725.6 Ferroresonance 765.7 Summary 776 Transformer Models - Applications 796.1 Introduction 796.2 Basic Transformer Equivalent Circuit 79V6.2.1 Single-Phase Two-Winding Transformer 806.2.2 Single-Phase Three-Winding Transformer 826.2.3 Three-Phase Transformer 836.3 Estimation of Transformer Parameters 866.3.1 Short-Circuit Tests 866.3.1.1 Single-Phase Transformers 866.3.1.2 Three-Phase Transformers 906.3.2 Open Circuit Tests 926.3.2.1 Stray Capacitances 946.4 Sensitivity Study 976.5 Field Test 986.6 Summary ioo7 Conclusions 102Refrrences 104A Orthogonality Between ‘r and I 110B Measurement ofthe InitialMagnetization Curve 112C Computation of Triplen Harmonic Components ii 4D RationalApproximation of the Real Part of Y(s) 116D. 1 Fitting Procedure 116viD.2 G(w) Obtained from Lamination Data 118D.3 G(w) Obtained from Standard Tests 120viiList of Tables2.1 Typical distribution transformer data 263.1 Laboratory measurements 393.2 Computed v - Ir and A - i curve 394.1 Three-phase transformer test data 555.1 Flux-current curve 715.2 Linear circuit parameters 746.1 Distribution transformers 876.2 Correction factors 946.3 Corrupted flux-current curves 966.4 Sensitivity study 97viiiList ofFiguresNonlinear element connected to linear networkSimultaneous solution of two equationsPiecewise linear inductance with two slopes“Switched inductance” implementation of two slope piecewise linear inductanceElementary domain hysteresis ioopLadder network for eddy current representationEddy current distribution in a ferromagnetic sheet of thickness dHysteresis Loops2. Typical steel lamination power loss curves2.10 AC core loss measurements2.11 Power loss curve for commercial grain oriented steel laminations2.12 Ferroresonance in a 1100kV test system2.13 Thêvenin equivalent circuit2.14 Nonlinear inductance characteristic referred to the 1100 kV side2.15 Voltage at phase A (sensitivity study for changes in core resistance)2.16 Voltage at phase A (sensitivity study due to variations in the air core inductance)2.17 Energization of a transformer2.18 Inrush current (sensitivity study)77881011141617191922222323242727ix3.1 Excitation test313.2 V,- Average power curve323.3 Computation of the nonlinear resistance 343.4 A-i1curve 403.5 Computed v - ir curve 403.6 Nonlinear resistance 413.7 Nonlinear inductance 423.8 Ferroresonance in a power system 423.9 Nonlinear core inductances 433.10 Nonlinear resistances 443.11 Newly manufactured transformer V,- I, curve 444.1 Delta-connected transformer positive sequence excitation test 484.2 Core representation 504.3 V,- power loss curve 524.4 Generating current waveform from sinusoidal flux 544.5 A - i curve 564.6 VirCUrve 575.1 Eddy current representation of the core after [18] 615.2 Eddy current representation of the core after [16] 61x5.3 Eddy current representation of the core after [20] 625.4 Core parameters 645.5 Realization ofRL networks 675.6 Frequency-dependent representation of the core 695.7 A - i hysteresis curve 705.8 Frequency-dependent resistance 715.9 Flux-current loops 725.10 Flux-current curve of a single-phase 1 kVA transformer 745.11 Transformer input voltage 755.12 Transformer inrush current 755.13 Transformer inrush current 765.14 Ferroresonance in a power system 776.1 Basic transformer model 806.2 Two-winding transformer 816.3 Frequency-dependent short-circuit impedance model 826.4 Three-winding transformer 836.5 Three-phase transformer short-circuit test 846.6 Impedance measured on the primary side 886.7 Impedance measured on the secondary side 886.8 Short-circuit impedances of single-phase transformers 88xl6.9 Measured and simulated L/R curves 896.10 Short-circuit impedance frequency response 906.11150 kVA three-phase transformer (self impedances) 916.12 500 kVA three-phase transformer (self impedances) 916.13 75 kVA short-circuit impedances (self impedances) 916.14 500 kVA transformer L/R curve 926.15 Single-Phase transformers 936.16 V,- I, characteristic for a 75 kVA three-phase transformer 946.17 Newly manufactured transformer 956.l8DCdrivesetup 986.19 Measured voltage waveform 996.20 Measured current waveform 996.21 Transformer secondary line voltage 99B. 1: Measurement of the initial magnetization curve 111B.2: Hysteresis loop locus 112D. 1: Frequency dependent resistance 117D.2 Transformer core loss curve at rated flux 120xiiAcknowledgementI would like to express my gratitude to those who helped me to complete thisresearch work. Special thanks are due to my supervisor, Dr. Hermann Dommel for hisencouragement, patience, guidance and for arranging much-needed financial support.I wish to thank Dr. José MartI for discussions and invaluable suggestions. I am grateful toDr. A. E. Araijo, Dr. S. Cameiro and Dr. A. Soudack for suggestions on the reseach work and onthe presentation on the results in this thesis.I am grateful to Dr. W. Dunford and Dr. M Wvong for their help at the early stage ofthiswork. Thanks are due to Mr. A.. Otter for providing financial help from TRIUIVIF (Tri-UniversityMeson Facility)-Vancouver, B.C., and for discussions and providing test material.Thanks are due to The Canadian Electiical Association (CEA) and Powertech Labs Inc., forfinancial assistance duiing part ofthis work and for providing transformer data from Project 267-D-766. The support of Mr. J. Drakos, Mr. M. B. Hughes and Mr. K. Takahashi, is gratefullyacknowledged. I am grateflul to Dr. Wilsun Xu for invaluable discussions, and for help on Chapter 6.The financial assistance from Universidade Federal da Paraiba (UFPB)-Campina GrandeBrazil, where I received the support ofthe Department ofElectrical Engineering, and from ConseihoNacional de Pesquisa (CNPq-Brazil), are gratefully acknowledged.I am deeply gratefhl to my wife Cataiina and my daughter Deborah for all their love,patience and help. I*iing this time, my wife brought light through our new born Eduardo, lookedafter the family, and kept encouraging me.Finally, to Kadi Pun-u, Claudia Lisbôa, Claudia Oliveira, Jane Armstrong and theirfamilies, the Soudack’s and those friends who cared, my warmest thanks.xliito:Elisa Silvaand,to the Memory ofEdison Guimaräes - “Padin”His early blindness by the sight did not prevent himfrom being a tireless learner of electricity and music.xivChapter 1Introduction1.1 Transformer ModellingThe simulation of electromagnetic transients in power systems is essential forinsulation coordination studies and for the adequate design of equipment and its protection.To carry out these studies on digital computers, mathematical models are needed for thevarious components, whether with lumped or distributed parameters. To attempt to modeleach component in its entirety and then to determine its interaction with the rest of thesystem would be extremely difficult due to the system complexities. Models with somesimplifications, which are still accurate enough for practical purposes, are therefore usuallyused.The problem of accurately predicting the transient electrical interaction of powersystem componnts has faced the electric power engineer for almost a century. There is alarge amount of research work in this area. The transformer is one of the most importantcomponents in power systems, and because of this, it has been given special attention [1].1Chapter 1. Introduction 2Despite the large number of papers published in the area, transformer modelling stillpresents substantial difficulties today. Transformer inductances are nonlinear and frequency-dependent. The distributed capacitances between turns, between winding segments andbetween winding and ground produce resonances that may affect terminal and internalvoltages [2]. The core modeffing may play a very important role for ferroresonance andinrush current studies in transformers [3].1.2 Thesis OutlineWhen power system transients are to be computed, general purpose programs suchas the EMTP (Electromagnetic Transients Program) are often used [4]. Our goal is toadvance the modelling of transformers in connection with these programs, focussing onsaturation, eddy currents and hysteresis effects in the iron core. The developed models areintended to be applicable for situations such as ferroresonance and inrush currents intransformers. The models are discussed as follows:• A brief literature review of transformer models is presented in Chapter 2.• Saturation in the core is represented by nonlinear functions obtained from thetransformer test data. The model development, measurements and simulations arediscussed in Chapter 3.• An algorithm to produce saturation curves of delta connected transformers, frompositive sequence open circuit tests, is developed in Chapter 4.• Frequency-dependent core models are presented in Chapter 5.• In Chapter 6, distribution transformer models are obtained from 60 Hz parameters.The difficulties in obtaining the transformer parameters are outlined. Comparisonsbetween simulations and a field test are carried out.Chapter 1. Introduction 3• The main conclusions ofthis thesis work are summarized in Chapter 7.1.3 Thesis ContributionsThe author believes the following to be original contributions from this research work:a) A direct method to more accurately compute saturation curves from transformerstandard test data (Chapter 3).b) An algorithm to compute saturation curves of three-phase delta-connectedtransformers in which the delta connection could not be opened for tests (Chapter 4).c) Development of frequency-dependent core models in which eddy current andhysteresis are treated simultaneously. For these models it is not necessary to pre-definethe trajectories ofthe dynamic hysteresis loops (Chapter 5).d) Guidelines to model distribution transformers from rated parameters (Chapter 6).The following publications report part of the research work developed in this thesis:1) W. L. A. Neves and H. W. Dommel, On Modelling Iron Core Nonlinearities, IEEETransactions on Power Systems, Vol. 8, No. 2, May 1993, pp.417-425.2) W. L. A. Neves, H. W. Dommel, Saturation Curves of Delta-ConnectedTransformers From Measurements, to appear in IEEE Transactions on PowerDelivery. Paper 94 SM 459-8 PWRD presented at IEEE PES Summer Meeting, July24-28, 1994, San Francisco, CA.3) W. L. A. Neves, H. W. Dommel and Wilsun Xu, Practical Distribution TransformerModelsfor Harmonic Studies, to appear in IEEE Transactions on Power Delivery.Paper 94 SM 406-9 PWRD presented at IEEE PBS Summer Meeting, July 24-28,1994, San Francisco, CA.Chapter 2Literature Review And Case Studies2.1 IntroductionA brief review of various transformer representations for digital simulation oftransients in power systems, and a discussion of eddy current and hysteresis loss in magneticcores are presented next. It is shown that an exact model which reproduces the frequency-dependent core losses, even at low frequencies, is very difficult to achieve. Sensitivityanalysis for ferroresonance and transformer inrush current case studies are carried out. Inthese studies, the transformer core is represented by a nonlinear inductance in parallel with aconstant resistance. It is shown that the system is more sensitive to variations in the coreinductance. The system is not sensitive to small variations in the shunt resistance. However,typical transformer data show that no-load losses at rated frequency increase quickly as thetransformer is driven into saturation. This may be significant for ferroresonance studies.2.2 EMTPBasic ModelsGuidelines to model transformers with the EMTP are presented in references [4,51.These models are based on circuit theory. The linear behaviour of transformers can be4Chapter 2. Literature Review and Case Studies 5represented by branch resistance and inductance matrices [R] and [LI (here the excitingcurrent must not be ignored since its absence produces infinite elements in the inductancematrix), or by a matrix [R] and an inverse inductance matrix [LI-1. These matrices areobtained from positive and zero sequence short circuit impedances and from open circuitimpedances. Saturation effects can be simulated by appending nonlinear inductancebranches. In [4], it is suggested that these nonlinear branches should be placed across thatbranch in the equivalent circuit where the integrated voltage is equal to the iron core flux.Although this point depends on the transformer design and, in general, is not accessible inthe model, it can be approximated fairly accurately by using the branch of the windingclosest to the core (usually the lower voltage winding). Saturation curves of transformersare often supplied as rms values of voltages and currents(V,,,—f(I)). A technique forconverting this curve to a peak flux versus peak current characteristic( A=f(i)) is suppliedby the auxiliary program CONVERT[4]. This algorithm does not take eddy currents andhysteresis losses into account, i. e., when computing saturation curves it is assumed that theexcitation branch consists only of a nonlinear inductance. The next two chapters presentimprovements on the computation of saturation curves by including the effect of transformerno-load losses.In the EMTP, nonlinear elements are either represented as piecewise linear or asnonlinear with the compensation method [6]. When the compensation method is used,nonlinear elements are simulated as current injections, which are superimposed on the linearnetwork solution without the nonlinear elements. As an example, consider a case where thenetwork contain only one nonlinear resistance between nodes k and m (Figure 2.1). Thenetwork solution is found by the compensation theorem according to the following steps:Chapter 2. Literature Review and Case Studies •6• remove the nonlinear branch between nodes k and m and calculate theopen circuit voltage v0;• build the instantaneous Thèvenin equivalent circuit between nodes k and m(to find the Thèvenin resistance, a cuffent of 1 A must be injected fromnode k, and drawn out from node m);• solve the two following equations simultaneously:v = v,,,0 —R(t). ‘km (2.1)= f(1icm) (2.2)Equation (2.2) represents the nonlinear resistance characteristic. Figure 2.2 shows thesimultaneous solution of the two equations above (intersection between the two curves). Fornonlinear inductances, the nonlinear characteristic is usually known in the form:(2.3)The EMTP uses the trapezoidal rule of integration and converts the flux A(t) into alinear ftinction of (t) and the network solution is found in a similar way as for a nonlinearresistance.The saturation characteristics of modern transformers are often represented aspiecewise linear inductances of two slopes (Figure 2.3). Such piecewise linear inductancescan be simulated with two linear inductances L1 and L2 in parallel (Figure 2.4). The switchis closed whenever Aj2SATURAT10N, and is opened again as soon asChapter 2. Literature Review and Case Studies 7linear partof networkk km-i?;1LkmFigure 2.1: Nonlinear element connected to linear network.VkmVkmOnonlinear resistancecurvenetwork curve‘kmFigure 2.2: Simultaneous solution of two equations.Chapter 2. Literature Review and Case Studies 8Figure 2.3 : Piecewise linear inductance with two slopes.1Figure 2.4: “Switched inductance” implementation of two slope piecewise lineainductance.2..‘SATURATIONL2‘SATUBA11ONkjVkmChapter 2. Literature Review and Case Studies 9Magnetic hysteresis effects have been incorporated in the BPA (Bonneville PowerAdministration) version of the EMTP [7]. This model uses pre-defined trajectories in theA — i plane to decide in which direction the curve will move if the flux either increases ordecreases. Eddy current effects in the core are represented as fixed resistances. Mork andRao [8] used this model to simulate ferroresonance and compared their results to laboratorymeasurements. There was a large discrepancy between measured and simulated curves. Asingle-valued flux-current characteristics predicted voltage and current waveforms in closeragreement to the tests. In Chapter 3, it is shown that a nonlinear resistance may be necessaryto represent eddy current effects in transformers.2.3 Other ModelsDick and Watson [9] described a method of saturating large power transformers andplotting instantaneous magnetization curves. The authors used a detailed equivalent circuittransformer model based on the principle of duality between magnetic and electric circuitswhich takes the yoke saturation into account. Hysteresis loops are modeled using predefined trajectories constructed from a hyperbolic equation.Germay et al. [101 studied ferroresonance effects in power systems. They representedmagnetic hysteresis by Preisach’s theory [11,12]. This theory assumes that the ferromagneticmaterial is made up of elementary domains and that the magnetization characteristic of eachdomain is a rectangular ioop characterized by the constants a and b (Figure 2.5), and by thedisplacement field Hm representing the action of neighboring domains. It also assumes that adistribution function, related to the probability of finding a ioop with given (a,b) is unique.The distribution function can be computed numerically by manipulation of the saturationloop and magnetization curve [11]. This theory has gained large acceptance. Its basic ideasChapter 2. Literature Review and Case Studies 10and evolution are presented by Mayergoys [12]. Recently, a hysteresis model based on thistheory was developed for the EMTP [13].Santesmases et al. [14] represent transformer cores by a simple equivalent circuitconsisting of a nonlinear inductance in parallel with a nonlinear resistance. The nonlinearelements are obtained from functions derived from the hysteresis dynamic ioops. This isessentially the same model as proposed by Chua and Stromsmoe [15]. The resistance in themodel accounts for the energy loss due to the loops, which means that the hysteresis andeddy current losses are assumed to have the same frequency dependence. A family ofdynamic hysteresis loops is needed to construct the nonlinear functions.MMa H;1HmFigure 2.5 : Elementary domain hysteresis loopA recent attempt to build a general transformer model for transient studies wassponsored by the EMTP Development Coordination Group (DCG) [16]. The principle ofduality was used to model the magnetic flux paths in the air and in the iron parts. Frequencydependent effects in the core were included by solving Maxwell’s equations (in theChapter 2. Literature Review and Case Studies 11frequency domain) within laminations, ignoring nonlinear effects1.As a result, a frequency-dependent equivalent impedance Zeq(W) was found. Zeq(W) was matched by a laddernetwork and connected in parallel with the iron core nonlinear inductance L1 as shown inFigure 2.6.Ri R3 R4The ladder network reproduces the theoretical transformer frequency response withan error less than 5% for frequencies below 200kHz. This model was applied in a situationwhere a circuit breaker, on the low voltage side of the transformer, attempts to clear a faultnearby [17]. The transient recovery voltage (TRV) is computed using both a frequency-dependent model for the core and the conventional model (constant resistance in parallelwith the magnetizing inductance). The conventional model produced a more damped TRy.2It is difficult to know which model is correct since the authors did not show comparisons tofield measurements.The idea ofmodelling power transformer eddy current effects by means ofMaxwell’sequations, has also been used in references [18,19,201. In the next section, some difficulties1 ferromagnetic material was assumed to have constant permeability t and constant resistivity a.2 In reference[17] it is also shown that the conventional model is accurate enough (numerical error lessthan 5%) for frequencies up to 3 kHz.R2Figure 2.6 Ladder network for eddy current representation.Chapter 2. Literature Review and Case Studies 12concerning classical eddy current and hysteresis representations of ferromagnetic materialswill be discussed.2.4 Iron LossesTransformer cores are usually made of iron alloys. Core materials can be divided intothree major classes: non-oriented steel (hot rolled)3,grain oriented steel (cold rolled)4 andmetallic glasses (amorphous material)5.Most distribution and power transformer cores in service today are of grain orientedsilicon steel laminations. However, given the very low losses of amorphous alloys, the trendmay change in the near future. Today, thousands of distribution transformers and a fewpower transformers made of metallic glasses are in service in the U.S.A., Japan and Canada[22].In the presence of a time-varying magnetic field, induced voltages, eddy currents andhysteresis take place in the core material. Classical electromagnetic theory assumes auniform distribution of eddy currents when slowly time-varying magnetic fields are appliedto iron cores (Figure 2.7a). The first theoretical studies of eddy currents in iron sheets weredone by Oliver Heaviside followed by J. J. Thomson [24]. The iron was assumed to be ahomogeneous medium characterized by two constants: permeability ji and conductivity a.Non-oriented grades of electrical sheets are designed to have the same magnetic properties in the rollingdirection as they have perpendicular to that direction. They were largely used in the past for power anddistribution transformer cores.rolled materials were introduced to the market in 1934 by N. P. Goss [21]. Their permeabilities aremuch bigger in the direction of rolling than perpendicular to that direction. Core laminations are usually cutso that the magnetic flux is along the rolling direction for the greatest part of its path through the core.These materials have lower losses when compared to non-oriented steels.5 alloys were introduced to the transformer market in the U.S.A. in 1976 [22]. These alloys presenthigher resistivity, when compared to grain oriented steels, and very low losses.Chapter 2. Literature Review and Case Studies 13The electromagnetic theory (Maxwell’s equations) was applied to show how the magneticflux density B would diffbse from one part of the material to the other according to theequationV2B= (2.4)The equation above oversimplifies a much more complicated phenomenon.6 Corelosses, computed by this approach, are always underestimated when compared to measuredvalues. A detailed review of eddy current and hysteresis loss in magnetic cores is presentedin [25]. Weiss, Barkhausen, Bitter, Landau and Liftshitz made significant contributionstowards the understanding of eddy current effects in ferromagnetic materials. In 1907, theFrench physicist Pierre Weiss provided the first insight into understanding the behavior ofmagnetic materials. He introduced the concept of magnetic domains. In 1919 Barkhausenhad shown that the magnetization could change in a very discontinuous way (Barkhauseneffect). in 1931, Francis Bitter, working at the Westinghouse Research Laboratories provedthe existence of domains by making them visible. His technique consists of polishing thesurface of the magnetic material and spreading a colloidal suspension of magnetic powderover the surface. The powder will be deposited in regions of higher gradient fields (domainboundaries) and the domains are then visible through a microscope[211. In 1935, Landauand Lifshitz introduced the ideas that magnetization could change by a movement of theboundary between domains, and that domains magnetized in the direction of the applied fieldwould expand at the expense of domains magnetized against the applied field. Today, it is6 Maxwell died in 1879, he left his theory in the form of twenty equations in twenty variables. Shortlyslier his death, the reduction of his equations to the four vectorial equations known today was doneindependently by Oliver Heaviside and Heinrich Hertz [23]. At that time, very little was known aboutferromagnetism. Eddy currents in ferromagnetic materials were assumed to behave the same way as in non-magnetic conductors.Chapter 2. Literature Review and Case Studies 14generally accepted that the eddy current loss is due to the micro eddy currents produced atthe moving domain boundaries. Therefore, eddy currents will be concentrated around themoving domain walls, as shown in Figure 2.Th. The bigger the domain the larger the eddycurrents produced around its boundaries. Eddy current distribution may not be uniform evenfor very slowly time-varying magnetic fields.(a)1;(b)Figure 2.7: Eddy current distribution in a ferromagnetic sheet of thickness d:(a) Classical representation;(b) Sheet subdivided into 180° domain of width a.LChapter 2. Literature Review and Case Studies 15It is very complicated to properly use Maxwell’s equations within iron corelaminations to account for eddy currents. For a precise calculation, the effect of domain wallmotion should somehow be included in the field equations [26, 27, ‘28].The loss in a ferromagnetic sheet at a frequency f consists of hysteresis and eddycurrents. The total loss is always greater than would be expected from the sum of statichysteresis loss and eddy current loss, calculated using classical theory [29,30]. The excessloss, arising from the non-uniform distribution of eddy currents, is known as anomalousloss.7 Since, in practice, only the total loss w can be measured, assumptions must be madebased on the most probable physical behavior of the material if hysteresis and eddy currentloss are to be separated. The total loss per kilogram, per volume ofmagnetic material, in aniron sample can be written as a combination of three loss components:WWh+We+Wa, (2.5)where Wh , We and Wa are hysteresis, eddy current and anomalous loss, respectively, inW/kg. It is well known that the total loss w is frequency-dependent. Hysteresis loss isattributed to domain wall movements back and forth across crystal grain boundaries, nonmagnetic inclusions and imperfections [211. It is common to assume that Wh is independentof the speed in which the domain wall moves. So, the hysteresis loss per magnetic volume ata given frequencyf is related to the enclosed area of the DC hysteresis loop (Figure 2.8)according to the equation:Wh=fHdB. (2.6)‘ Electrical Engineering textbooks, usually address the magnetic domain theory to explain the properties ofmagnetic materials, but seldom relate the domain wall movements to eddy currents. Anomalous losses wereknown even before the domain theory was completed. In 1927, it was already known that there is a strongcorrelation between grain size and eddy current losses [31].Chapter 2. Literature Review and Case Studies 16HFigure 2.8: Hysteresis loops. In the B - H plane, the trajectoryof a signal of frequency f, will encircle the hatchedarea (DC hysteresis loss) f times per second.Any increase in loss per cycle above the DC hysteresis loss has been attributed to eddycurrent effects [22]. Steinmetz [28] proposed the following equation for calculation of thehysteresis loss:WhkhB&f, (2.7)wherekh is the hysteresis coefficient and depends on the core material;Bm is the maximum flux density in Teslas; andx is the Steinmetz coefficient (ranging from 1.5 to 2.2 depending on the core material).The principal means of controlling the core loss is to use thin laminations. For alamination in which its thickness Ia is much smaller than its width, the classical eddy currentloss is given by [22]:BIWe(2r d Bm .f)2/(6p), (2.8)Chapter 2. Literature Review and Case Studies 17where p is the resistivity of the material, and Id is the lamination thickness. Insertingequations (2.7) and (2.8) into (2.5) and dividing by f, the total loss per cycle is given by:=kh BmX +(2td Bm)2fI(6p)+Wa IfAtypical curve of power loss per cycle as a fianction of frequency, for a constant fluxamplitude, is shown in Figure 2.9. The anomalous loss can be very high (usually greaterthan the classical eddy loss for commercial steel at power frequency [261).a)C.)C.)G)0U)U)0a00FrequencyFigure 2.9: Typical steel laminations power-loss curve.Chapter 2. Literature Review and Case Studies 182.4.1 Laboratory MeasurementsAs part of this thesis project, some measurements were perfbrmed for grain orientedsteel laminations to gain some insight into how eddy currents and hysteresis loss behave asthe frequency changes. The steel samples were assembled in a standard Epstein frame [33]. /The amplitude Bm of the sinusoidal flux density B =Bm sinwt was kept constant during eachset ofmeasurements.Figure 2.10 illustrates the circuit used to measure the total AC core loss for afrequency range from a few hertz up to 80 Hz. V(t) is a frequency variable sinusoidalvoltage source connected to a power amplifier. A waveform analyzer was used to measurethe voltages at points A and B with respect to ground. The current sample waveform wastaken from a 0. 12 resistance R connected in series with the Epstein frame primary winding.Current and voltage waveforms (512 points) were obtained and the losses were computedusing a built-in routine. The total loss per cycle, as a function of frequency for Bm =1.OT, isshown in Figure 2.11. The laminations were 0.3mm thick, with p = 4.5 x i02. m. The solidline, through the measured points, is a second order polynomial approximation. This curveis extended downwards to f=0. At f0, it is assumed that the loss per cycle is the DChysteresis loss. The classical eddy curent loss is computed using (2.8) and added to thehysteresis loss. The losses per cycle calculated by the classical approach are lower than themeasured ones. The hysteresis loss per cycle at 60 Hz accounts for about half of the totalloss. Reference [32] quotes measurements in grain oriented steel laminations in which theanomalous loss could be close to an order of magnitude higher than the classical eddycurrent losses for frequencies up to 1 kHz. Herzer and Hilzinger [34] show examples ofamorphous alloys with large anomalous loss (nealy 40% of total loss) at frequencies of 100kHz.Chapter 2. Literature Review and Case Studies 19It is complicated to predict losses in iron cores accurately. A better understanding ofloss mechanism in ferromagnetic materials is providing researchers with the tools to reducethem. Theoretically, the classical methods would be applied properly if the laminations weremade of a fine domain structure. Researchers are struggling to reduce the total loss andincrease permeability of ferromagnetic steels. Very high permeability low loss steel sheetsare on the market today. Nevertheless, although total losses are low, anomalous loss is stillhigh when compared to the classical eddy current loss [221.V(t) (EZEEEEJEFigure 2.10 : AC core loss measurements.10-‘ 80)-3E6ci)C.)>—0Cl)U)0.J 220 40 60 80 10Frequency (Hz)Figure 2.11: Power-loss curve for commercial Grain oriented steel lamination.Measured LossesClassical LossesChapter 2. Literature Review and Case Studies 202.4.2 Building FactorThere is also a further complication that transformer core loss per kilogram is alwaysgreater than the nominal loss of the steel as measured in standard testers. The ratio betweenthe transformer per unit loss and the nominal or standard unit loss is called the “buildingfactor” of the core. Building factors usually range from 1.1 to 2.0 [35]. The extra loss is dueto phenomena such as:• Non-uniform flux distribution due to difference in path lengths among magneticcircuits;• Distortion of flux waveform due to magnetic saturation;• Flux directed out of the rolling direction;• Transverse flux between layers due to joints.The flux distribution in transformer laminations is not uniform even at lowfrequencies [36,37,38]. For a sinusoidal applied flux, the flux in each lamination is notsinusoidal, although the flux components add up to produce the sinusoidal total flux.Advances in computer software have been used to improve the design of electricalmachinery [39]. There are several commercial programs available today [40]. They areessentially usefhl for situations in which qualitative results are important (for instance, indesigning transformer lap joints, it is important to find the geometry of the joints that leadsto minimum losses). The accuracy of the present methods needs verification againstexperiments [35].Chapter 2. Literature Review and Case Studies 212.5 Transformer Core Representation During TransientsThe major nonlinear effects in transformers are saturation, eddy currents andhysteresis. Saturation is the predominant effect [41]. In the following sections,ferroresonance and inrush current simulations will be addressed. The transformer excitation /branch is represented by a crude model (constant resistance in parallel with a nonlinearinductance). Sensitivity studies are carried out to analyze how the system responds tochanges in the core model parameters.2.5.1 Ferroresonance Case StudyConsider the BPA (Bonneville Power Administration) 1100kV test system [42]. Itcomprises a generating station, a transformer bank (autotransformers) and a short three-phase transmission line. Field tests were carried out. Ferroresonance occurred in phase Awhen this phase was switched off on the low voltage side of the transformer (Figure 2.12).Phase C was not yet connected to the transformer at that time. One can study this case,replacing the dotted part of the network by its Thèvenin equivalent circuit, which consists ofa voltage source behind a capacitance (crude representation of the capacitive coupling tophase A of the line). Figure 2.13 is the equivalent circuit of the system referred to the highvoltage side. The nonlinear inductance characteristic shown in Figure 2.14 (three straightline segments) was obtained from the curve supplied by the transformermanufacturer, using the method of [4]. Unfortunately, the transformer no-load data wereproduced by exciting voltages that did not go beyond 1.1 p.u., and data at higher saturationlevels would be needed for this case. Autotransformers have typical air core inductances (thecore is completely saturated and it behaves like air) of 3 to 4 times the short circuitinductance [4]. A straight line segment, with a slope of 4 times the short circuit inductancewas connnected to the last segment of Figure 2.14, to represent the air core inductance. TheChapter 2. Literature Review and Case Studies 22Microtran® program[43J was run twice for different values of core resistance (RRc andR0.8R, where Rc=4.2M2 is the resistance at the rated voltage). A time step of zt=1OOiswas used in each case. Both simulations, of the terminal voltage a’t phase A, are shown inFigure 2.15. The two curves are almost identical.Figure 2.12: Ferroresonance in a 1100kV test system.152 Q 11.3f2 742L2 7422 11.3L2 O.O2619F635. 1/Q.kV. 131.0 Z12 kVFigure 2.13 : Thèvenin equivalent circuit.Chapter 2. Literature Review and Case Studies 23(0>x:53000200010002 4 6 8Current (A)Figure 2.14 : Nonlinear inductance characteristic referred to the 1100 kV side.1200800400I0-400-800Time (ms)Figure 2.15: Voltage at phase A (sensitivity study for changes in core resistance).RRcR=0.8Rc20 40 60 80 100 120 14Chapter 2. Literature Review and Case Studies 24The Microtran program was run again, now with the slope of the air core inductancemade 20% smaller. The results are shown in Figure 2.16, where the solid curve is the samecurve of Figure 2.15 for R=R. It is seen that the simulated sytem is more sensitive tovariations in the nonlinear inductance than to core losses.Correct air core inductance1200 — Air core inductance 20% smaller800C400-400—-800 —- I I ‘ I ‘ I20 40 60 80 100 120 14Time (ms)Figtue 2.16: Voltage at phaseA(sexivity studydue to vaiiations hthe air core inductance).2.5.2 Inrush Current Case StudyNow consider the analysis of inrush current simulations in a single phase transformer(same transformer data of the previous section). The aim here is to see how sensitive thecurrent waveforms are to changes in the core resistance and nonlinear inductance (FigureChapter 2. Literature Review and Case Studies 252.17). The plot of the inrush current for three values of core resistance (R=R, R0.5Rand R=oo) and the nonlinear inductance of the previous example, is shown in Figure 2.18.The solid curve represents in fact any of the simulations. It was asumed that no transformerresidual flux existed prior to energization at t0. In these simulations, inrush currents are notsensitive to variations of the core resistance. However, if the slopes of the flux-currentcharacteristic above the rated flux were changed by 10% (dashed curve), a noticeabledifference between the current waveforms would be seen. In these simulations, inrushcurrents are shown to be very sensitive to variations of the nonlinear inductance.2.6 SummaiyThis chapter summarized the difficulties in modelling transformer cores. Their exactrepresentation during transients is complex since the magnetic properties of the cores are notyet fully understood. Another major problem is the availability of data. All that is usuallyavailable from transformer manufacturers are data obtained from tests performed at ratedfrequency [46,47,48]. The information available from open circuit tests are rms voltages as afunction of rms currents, and no-load losses for a few input voltage levels.In the simulation examples of Section 2.4, a constant resistance was used toreproduce the transformer core loss. It was shown that the system is not sensitive to smallvariations in the resistance. In practice, however, this resistance is not linear. Typicaldistribution transformer correction factors are shown in Table 2.1 [44]. They should beapplied to no-load losses, at rated voltage, to give the correct losses when the transformer isdriven into saturation. The equivalent resistance, which reproduces the open-circuit losses,decreases as the voltage level increases beyond the rated voltage (losses increase at fasterrate than the square of the voltage). For 225 kVA and above (three-phase transformers), theopen circuit equivalent resistance is equal to approximately half of the resistance at ratedChapter 2. Literature Review and Case Studies 26voltage. This may make a difference during ferroresonance studies. Another complication isthat the core loss is frequency-dependent [29,38,49].Table 2.1 - Typical distribution transformer data.Operating CorrectionVoltage (%) Factors105 1.15 For 167 kVA and below, 1 phase110 1.30 For 150 kVA and below, 3 phase105 1.5 For 225 kVA and above, 1 phase110 2.4 For225kVAandabove,3phaseChapter 2. Literature Review and Case Studies 27635. 1LLkVFigure 2.17: Energization of a transformer.Correct air core inductance150 — Air core inductance 10% smaller100 —50-20 40 60 80 10Time (ms)Figure 2.18: Inrush current (sensitivity study).Chapter 3On Modelling Iron Core Nonlinearities3.1 IntroductionTransformer manufacturers usually supply saturation curves in the form of rmsvoltages as a function of rms currents. Some methods have been used to convert these—I,, curves into peak flux - peak current curves (nonlinear inductances) [4, 45, 511.As shown later, these methods can be modified to take iron core losses into account, therebyproducing a nonlinear inductance as well as a parallel nonlinear resistance.In addition, laboratory experiments were performed with a silicon iron steel coreassembled in an Epstein frame. Average power and rms current at 60 Hz were measured atdifferent voltage input levels. For comparison purposes, the initial magnetization curve forthe core material was measured as well.Simulations of ferroresonance in a power system are carried out to examine theeffect of the transformer nonlinearities on its terminal voltage waveform. These simulationsare compared to a field test.28Chapter 3. On Modelling Iron Core Nonlinearities 29The algorithm is applied to distribution transformers to get their open circuitnonlinear parameters. Measurements have shown that there are newly manufacturedtransformers in which their excitation currents at rated voltag’e are smaller than theirexcitation currents at lower voltages. For these transformers, improvement on the computed /saturation curves would be accomplished by modif,’ing the algorithm to include the effect ofstray capacitances.3.2 Saturation CurvesThe cores of transformers and reactors are sometimes represented as an equivalentcircuit consisting of a nonlinear inductance (A—ii curve) in parallel with a nonlinearresistance (v— ‘r curve) [14, 15]. The characteristics of these elements are obtained from thedynamic hysteresis loops. The resistance in this model accounts for the energy losses due tothe loops. Chua and Stromsmoe [15] did make comparisons between simulations andlaboratory tests for a small audio transformer, and for a supermalloy core inductor as well. Afamily of peak flux - peak current ioops for 60, 120 and 180 Hz sinusoidal (voltages andcurrent) excitations of various amplitudes were obtained. The agreement betweensimulations and measurements of the loops was very good. This indicates that, for thefrequencies under consideration, a nonlinear resistance would represent hysteresis and eddycurrent effects reasonably well.The same equivalent circuit is used here. However, the nonlinear characteristics arecalculated in a simpler way directly from the transformer test data. The nonlinear resistance(piecewise linear v—i,. curve) is found from the no-load (excitation) losses. This informationis then used to compute the current through the nonlinear inductance and to construct thepiecewise linear A — I, curve.Chapter 3. On Modelling Iron Core Nonlinearities 30Figure 3.1(a) shows a voltage source connected to a single-phase transformerwhose excitation branch is represented by a nonlinear inductance in parallel with a nonlinearresistance. Their nonlinear characteristics are computed according to the followingassumptions:• the Vr and ?—i, curves (Figures 3.1(b) and 3.1(c)) are symmetric with respectto the origin (Rk and Lk are the slopes of segment k of the VIr and A—i1 curves,respectively);• the no-load test is performed with a sinusoidal voltage source; the windingresistances and leakage inductances are ignored.The conversion algorithm works as follows:For the construction of the VIr curve (Section 3.2.1):• compute the peak values of the current Ir(t) point by point from the no-loadlosses, and subsequently compute their rms valuesFor the construction of the A—i, (Section 3.2.2):• obtain the rms values Il.-rmj of the current i, (t) through the nonlinear inductancefrom the total rms I,,.. current and the applied voltage v(t);• compute the peak values of the inductive current i,(t) point by point from their rmsvalues and rms voltages.Chapter 3. On Modelling Iron Core Nonlinearities 31v(t)Figure 3.1: Excitation test:(a) core representation;(b) V— ‘r characteristic;(c) ). — i characteristic(a)V(b) (c)Chapter 3. On Modeffing Iron Core Nonlmearities 323.2.1 Computation ofthe v - Ir CurveLet us assume that the no-load losses I, F,..., m are ava,ilable as a function of theapplied voltage , ..., V as shown in Figure 3.2.ms1 Vrrns2m3Figure 3.2 : Vrms - Average Power curveFrom these data points we want to construct a piecewise linear resistance curve, as shown inFigure 3.3(b), which would produce these voltage dependent no-load losses. Let us firstexplain how the no-load losses can be obtained from a given v—i,. curve, before describingthe reverse problem of constructing the v—i curve from the given no-load losses at ratedfrequency. For instance, assume that the applied voltage is and varies sinusoidally as afunction of time, as shown in Figure 3.3(a), withv2(O)=VsinO (3.1)where 2 =V, Because of the symmetry of the v — 1 curve with respect to the origin, itis sufficient to observe 1/4 of a cycle, to 0 = ir /2. From Figure 3.3, it can be seen that:Chapter 3. On Modelling Iron Core Nonlinearities 33(V sin 0)1R1 if 0<01‘r(0)ji +(V sin0—V1)/R2 if 01In general, ‘r (0) can be found for each v(O) through the nonlinear v — r characteristic, eithergraphically (as indicated by the dotted lines in Figure 3.3), or with equations. This will giveus the curve i,-(0) over 1/4 of a cycle, from which the no-load losses are found as. =_$V(0)i(0)d0 (3.2)Let us now address the reverse problem, i.e., constructing the v—i,. curve from the given no-load losses. Obtaining the points,V,..., Vm on the vertical axis of Figure 3.3(a) issimply a re-scaling procedure from mis peak values,(3.3)for k = 1,2,3, ..., m. For the first linear segment in the v ‘r curve, the calculation of thepeak current 1,-i, on the horizontal axis is straightforward. Since P1=VI,,,,, in the linearcase,21ri-v1.v(e)V3V2ViChapter 3. On Modelling Iron Core Nonlinearities 34Figure 3.3: Computation of the nonlinear resistance:(a) sinusoidal voltage input signal;(b) Vr curve to be computed;IV(a)01It20 (c)(c) output current.Chapter 3. On Modelling Iron Core Nonlinearities 35For the following segments (k >2), we must use the power definition of equation (3.2), withthe applied voltage v(0) = 17k sin 0 (Figure 3.3a). Then=[s:1VkSiflo(0Jdo +f8 (J’ sin 0Irj + SO— ]dO+ + (3.5)( +sin o-The “break points” 01, 02, ..., 0k—I in equation (3.5) are known from0 =arcsin(VJIVk), (3.6)for j = 1, 2, ..., k — . The only unknown in equation (3.5) is the slope Rk in the lastsegment. The average power can therefore be rewritten in the formbrPk=ari+—-, (3.7)with ark, brk and Pk known values. Rk is then easily computed and In is calculated from-r Vk-Vkllrklrk_I+AkThis computation is done segment by segment, starting with ‘n2 and ending with thelast point ira. Whenever a point Irk has been found for the horizontal axis in Figure 3 .3b, itsrms value is calculated as well, because it is needed later for the construction of the ? —curve. 4, is found from the definition of the rms value,Chapter 3. On Modelling Iron Core Nonlinearities 36I2_Ji (o)do (3.9)i.e.,=[J:f0 dO+192( VksinOJIj jr + dO+ ... + (3.10)Ji [ir*i+ Vk Sifl O_do].3.2.2 Computation ofthe A - i, curveThe A — curve is computcd using the rms current information from the v—I,. curve.Peak voltages are converted to peak fluxes and the rms values of the current through thenonlinear inductance are converted to peak values.The conversion of peak voltages Vk to peak fluxes Ak is a re-scaling procedure.Hence, for each linear segment in the A—i1 curve,(3.11)where Co is the angular frequency.Let us now compute the peak values of the inductive current. At first, their rmsvalues are evaluated. It can be shown that for sinusoidal input voltages, the harmoniccomponents of the resistive current are orthogonal to their respective harmonic componentsof the inductive current (see Appendix A). Then,Chapter 3. On Modelling Iron Core Nonlinearities 37Il_nnsJItrnL_Ir2_rm5, (3.12)with the resistive current Ir_ already computed from equation (3.10) and the totalcurrent known from the transformer test data. For the first linear segment in the 2k—i,curve,= 1i-rmsi (3.13)For the following segments k 0, the peak currents are obtained by evaluating I,_ foreach segment k, using equation (3.9). Thus, assuming Ak(O)=Aksin8, we have1=[s:i[0 dO +SQZ[I ÷ Aks1n0_?jdO+... + (3.14)+sin 0— k-1 j do]Here, similarly to the case of the v — I,. curve computation, only the last segment Lk ofequation (3.14) is unknown. Equation (3.14) can be rewritten in the formalkrk+bIj’k+clk =0 (3.15)1For computation of the rms value of the inductive current, it does not matter what the flux phase is, owingto the fact that the voltage (or flux) is assumed to be sinusoidal and the 2 — curve symmetric with respectto the origin. Here, for computing purposes only, it is assumed (0) = Ak sin8. This has the advantagethat the limits of integration in equation (3.14) are the same as those in equation (3.5). The same procedureapplied in Figure 3.3 for the computation of the v— ir curve can then be used for the A. — i, curvecomputation.Chapter 3. On Modelling Iron Core Nonlinearities 38with constants a,, b,k and c, known, and rk =1/Lk to be computed. It can be shown thata4 > 0, 1i4 > 0 and Cik <0. Since 1 must be positive, then— —b,k +,,Jb, —4a, Cikr’k— . (3.16)The peak current 14 is computed from‘1* J*-i +f’k(Ak —Ak_I).In this fashion, the peak values of the inductive current are computed directly for everysegment in the A—i, curve.3.3 Comparisons Between Experiments and SimulationsLaboratory experiments were performed with a silicon iron steel core assembled inan Epstein frame [33]. No-load losses and rms current at 60 Hz, were measured fordifferent voltage levels (Table 3.1). For comparison purposes, the initial magnetization curve[50] for the core material, was measured as well (Appendix B). The computed v—i,. andA — i, points (including core losses) are shown in Table 3.2. The measured and the calculatedpoints (connected by straight line segments), with and without including the core losses, areshown in Figure 3•42 The computed v—i,. points connected by straight line segments (thefirst two columns of Table 3.2) is shown in Figure 3.5.It can be seen that the computed A — i, curve is closer to the measured one if weconsider the core losses. The V—I,. curve (Figure 3.5) is nonlinear and this may be importantwhen modelling transformers and reactors for transients or harmonic studies.2 Sometimes, due to measurement errors, V -I and no-load loss curves may be crooked and need to besmoothed. The developed algorithm checks the presence of “noise” and, if it exists, a low pass Fourier filteris used to remove the “noise” from the input data.CD C CD -.4.CDCD 0 -t 0 CD CD.---1>PP3D40.000G0o00Dcoo0c000000000000Q0‘4)—‘—00000000000000c’-’PPPpOpppopop•000000000000000O’0-0000.4)00—ppppppppppp.0M.t00000.D00c.’i0.-‘0.-—CI.t4)\O00----0cooc0)00Q’00Q00ON0000‘.D40‘000000000o00000p0p0pcppV1I-,Jt)•00000—aQN400j-oON(Ji•ON000.,--t4)—000000I-ON00000-‘0I..)ON-0ci0)0—0000000 0-CDChapter 3. On Modelling Iron Core Nonlinearities 400.10— ____e0 fAOv.vo —> /S..—...ci0) 0.06- ./ 0 Losses included0.04 1 A Losses not Included0.02- f • Measured pointsI I I0.20 0.40 0.60 0.80 1.00Current (A)Figure 3.4: A. —i1 curve.40.00—30.00 ——.5>20.00—0o ->10.00 —1111110.05 0.10 0.15 0.20 0.25Current (A)Figure 3.5: Computed v — Er curve.Chapter 3. On Modelling Iron Core Nonlinearities 413.4 Ferroresonance Simulations and Field TestThe BPA System of Section 2.4.1 is simulated again, Now, the transformerexcitation branch is represented by a nonlinear resistance (curve represented by threepiecewise-linear segments in Figure 3.6) in parallel with a nonlinear inductance (Figure 3.7)produced by the described algorithm, from the transformer manufacturer data. A straightline segment, with a slope of 4 times the short circuit inductance was connnected to the lastsegment of Figure 3.7 to represent the air core inductance. The voltage waveforms at thetransformer terminal at phase A line side, are shown in Figure 3.8. The simulations weremade assuming the excitation resistance to be constant (Rc=4.2M2 - see dotted curve) andassuming a nonlinear resistance represented by Figure 3.6 obtained from the algorithmdeveloped in this chapter (thin solid line). Simulations come closer to the field test (thickersolid line) if the nonlinear losses are taken into account.1000800>6000)z 4000>2000.40Current (A)Figure 3.6: Nonlinear resistance.3 field test curve was obtained using a digitizer to copy the data points from an oscilograph plot.Simulations were also made using a transformer air core inductance of 3 times the short circuit inductance,but the agreement with the field test was not so good.0.10 0.20 0.30Chapter 3. On Modelling Iron Core Nonlmeanties 423000 —Current (A)Figure 3.7 Nonlinear inductanceField TestConstant Resistance1200Nonlinear Resistance::(10-400-800 ,I I’ I ‘ I20 40 60 80 100 120 14Time (ms)Figure 3.8 Ferroresonance in a power system.Chapter 3. On Modelling Iron Core Nonlinearities 433.5 Distribution Transformer Saturation CurvesIn Figure 3.9, one can see the nonlinear inductances curves (2—i curves) for 3distribution transformers (50 kVA each), from different manufacturers, computed with thedeveloped algorithm from tests performed at the low voltage terminals (120 V). The no-loadlosses were measured as well. The nonlinear resistance curves are shown in Figure 3.10.Transformer A has higher inductance in the unsaturated region and saturates at higher fluxlevel, also it has the lowest no-load losses.The ,—I,,,,.. curve for a brand new distribution transformer is shown in Figure 3.11.The excitation rms current is not monotonic and drops as the voltage increases up to thevicinity of the rated voltage. In the unsaturated region, the transformer core has such a highpermeability that current through stray capacitances tend to cancel out the magnetizingcurrent.0.6C,)>0.40.2ABC5 10 15 20 25Current (A)Figure 3.9: Nonlinear core inductances.Chapter 3. On Modelling Iron Core Nonlinearities 44200150 ..•‘..— —— —/ ,- ——‘ / / 7/ ////ioo- s 7/’ A///BCurrent (A)Figure 3.10: Nonlinear resistances.150 —>100-0)7)-I0>50-2 4rms Current (A)Figure 3.11: Newly manufactured transformer —J curve.Chapter 3. On Modelling Iron Core Nonlinearities 45The developed algorithm will not work for this case, unless stray capacitances areknown. The algorithm is modified by inserting —WCVksinO, where C is the open circuitcapacitance, in between each parenthesis of equation (3.14). Then,IiWk =[s:1.nb_w Cvsinoj dO +J82[I + Ak0— CVk sinOdO+ ... + (3.17)f[1k1+ smn— a) CV sin oj do].Equation 3.17 is rewritten in the form of 3.15, and the inductance Lk computed foreach segment k. In Chapter 6, a crude method of estimating the capacitance C is brieflydescribed.3.6 SummaryA direct method for the computation of iron core saturation curve (A — I,) has beenpresented. It is based on the transformer test data. It is a modification of previous methods,with core losses taken into account. Besides the A—i, curve, it produces a nonlinear vcurve as well. Comparisons between laboratory measurements and simulations were made.It was shown that more accurate A — i, curves can be obtained if losses are included.Ferroresonance simulations were carried out. Simulations come closer to field tests ifthe nonlinear v — i,. curve is taken into account.Chapter 3. On Modelling Iron Core Nonlinearities 46The — i, and v— ‘r curves can be used for modelling transformers and iron corereactors in electromagnetic transients and harmonic loadflow programs.Chapter 4Saturation Curves ofDelta-Connected TransformersfromMeasurements4.1 IntroductionIn the algorithm presented in last chapter (Section 3.2), we assumed all odd harmonic current components to be present in the measured values. For three-phasetransformers, the standard excitation test data available are the positive sequence Vrrns -Irms curves, and no-load losses. In Figure 4.1, we show a symmetrical three-phase voltagesource supplying a no-load delta-connected transformer. The delta branches consist of nonlinear elements. In general, excitation tests are carried out with a closed delta [52]. In thatcase, ammeters, placed in series with the line, will not “detect” the triplen harmonic currents,because these circulate in the delta connection. In the next sections, we develop a methodfor generating the piecewise linear saturation curves (nonlinear resistance and nonlinearinductance), which accounts for the fact that triplen harmonics circulate in the closed delta,but do not appear in the measured line currents.47Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 48Figure 4.1: Delta-connected transformerpositive sequence excitation test.4.2 Basic ConsiderationsIn the circuit of Figure 4.1, the three branch elements of the delta connection areassumed to be nonlinear and identical. The branch currents can be written as a Fourier timeseries containing odd harmonic components only. Then:Iab(t) = I sin(o t)+ 13 sin(3 o.J t)+... + Ii,, sin(p Co t)+...ibC(t) = I sin(oit—120°)+13sin(3(cot—12O°))+”.+I sin(p(wt—120°))-i-... (4.1)Ica(t)= J sin(ot+120°)+1 sin(3(Cot+120°))+” +I, sin(p(w t+120°))+”.,where p is odd.The triplen harmonic currents (13, 19, ...) are in phase (zero sequence harmonics).Therms current in each branch is___________________/12÷12++12+i(t)‘A-rtnsj 2(4.2)Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 49The line currents are:‘a (t) 1o,(t)Ica(t)ib(t)i(t)—i(t) (4.3)i0(t)=Ica(t)4,.(t)Substituting (4.1) into (4.3) gives:1a(t)=S11(0)t3O°)+[5fl(50)t+30°)+”ibQ)=’f1l sin(wt —15O°)+’II5 sin(5wt+150°)+•• (4.4)sin(t+ 90°)+ sin(5ofl _900)+..Ammeters, placed into the supply line (outside the delta) read the rms current,112+ 2+ 2i-II’ pInns VFrom equations (4.1) to (4.5) one can make the following observations:• triplen harmonic currents, although present in each branch, are not present in theline currents.• if triplen harmonic currents in each delta branch are removed from the mis value(equation (4.2)) and scaled by 1, the rms line currents (equation (4.5)) areobtained. This is the basis of the algorithm developed next.4.3 Saturation CurvesEach delta branch in Figure 4.1 is represented by a nonlinear inductance in parallelwith a nonlinear resistance (Figure 4.2). Their nonlinear characteristics are computed withthe same assumptions made in the previous chapter.’‘Branch currents in the delta will have a “1k” subscript and line currents will have no subscript added. Forexample, ir4 and ‘r are the resistive components of the current in the delta branch and in the line,respectively.Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 50(b)a(a)jca(t)(c)Figure 4.2: Core representation:a) Nonlinear elements;b) VlrA curve;c) A.—i curve.itCVChapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 51The algorithm works as follows:1. For the construction of the v—4A curve (Section 4.3.1):• compute the peak values of the branch current 4, i,... point by point fromthe no-load losses.2. For the construction of the A—ij curve (Section 4.3.2):• from the v—i,,. curve, compute the rms values ‘ri, remove the triplenharmonic currents and obtain the resistive line current Ir-r,,,;• obtain the rms values I,_ of the line current due to the nonlinear inductancefrom i,., the total line current and the applied voltage v;• compute the peak values of the inductive current ,-A2 point by pointiteratively.4.3.1 Computation ofthe v - irA curveSimilarly to the previous chapter, let us assume that the three-phase no-load lossesJ, 1, . .., F,, are available as a function of the branch voltages ,, V, . .., V(Figure 4.3).Ifwe assume that the applied voltage is sinusoidal, the conversion of rms voltages topeak values (vertical axis ofFigure 4.2b) is simplykVrmsk%J (4.6)for k1,2,...,m.Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 52PP3P2P1VFigure 4.3: ,- power loss curve.Due to symmetry reasons, voltage and current waveforms need only be evaluatedover 1/4 of a cycle. For a sinusoidal voltage v(O) =Vk sin 0, the three-phase active power ]can be written in the form2:F)=3 [JV(O)rA(O)dO) (4.7)For the first linear segment in the v—i, curve, the current is sinusoidal. Thecomputation of the first peak current ‘rtSl is therefore straightforward. Since P1 = 3V1I inthe linear case,(4.8)From the second segment onwards (k2), equation (4.7) is evaluated at eachsegment k, with only i being unknown, as explained in more detail in Section 3.2.1 of theprevious chaptçi. The computation of the peak current is done segment by segment,If the voltages have no harmonics, the active power Pk is produced only by the fundamental component ofthe current. This component is present in each branch as well as in each line. So, it does not matter if thewattmeters are connected in series with the line or with each delta branch. The three-phase power readingswould be the same in both cases.\/rmsi V’r,ns Vrms3Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 53starting with ‘rA2 and ending with the last point i,. Whenever a point i is found, its rmsvalue is calculated as well. A Fourier program (see Appendix C for the algorithm) is used tocompute the triplen harmonics (13, 4, ...). They are then removed from to obtain‘r-rms, which is needed later for the construction of the 2—E, curve.4.3.2 Computation ofthe 2 - i,4 curveThe conversion of the peak branch voltages k to flux Ask, is again a re-scalingprocedure. Hence, for each linear segmcnt in the 2—i curve,(4.9)Let us now compute the peak values of the currents i through the nonlinearinductance. First, the rms values of the line currents 4. are evaluated withI,_, =4(It_)2_(1)2 (4.10)where the line current I_ is available from the measurements, and where ‘rl1L has alreadybeen computed from the previous section.For the first linear segment, the computation of ii is straightforward since there areno harmonics yet. Therefore,qf5I,_From the second segment onwards (k2), the algorithm works iteratively as follows(see A.—i,, curve in Figure 4.4):1. guessig;2. with A(O) = 2”k sin 0, find 1/4 of a cycle of the distorted current analytically;Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 543. compute the rms inductive branch current whose peak is ig;4. use a Fourier program to find the triplen harmonic inductive currents in the deltabranch (Appendix C);5. remove the triplen harmonics from the estimated rms branch current. Scale theestimated result ‘lest by and compare it to Iltms in equation (4.10);6. if the absolute value of the difference IIlr,ns - ‘lest is less than the specified tolerance, convergence is achieved. Otherwise, the residue 1d is added to ig andthe iterative process is repeated from step 2 onwards.0Figure 4.4: Generating current waveforfrom sinusoidal flux.correct curveIt201It21 0’ iz and initial guess/(Ii_rms& 1—nnsk1)/for every k 2. In some cases more than 20 iteration steps may be necessary.4.4 Case StudyConsider a 50 Hz three-phase five-legged core type transformer. The followinginformation is known [53]:1. rated power - 750 MVA (three-phase);2. rated voltages - 420 kV/ 27 kV (line to line values);3. wye connection on 420 kV side, delta connection on 27 kV side.The positive sequence excitation test data, from the closed delta 27 kV side, areshown in Table 4.1.V, is the rms line to line excitation voltage, is the rms excitation current(three-phase average) and P are the no-load losses (three-phase values).Table 4.1: Three-phase transformer test dataChapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 55In general, convergence is achieved in less than 20 iteration steps with a tolerance of‘g =l1_j +Vr(kV) (A) (kW)22.76 8.20 206.2124.29 11.35 240.2625.64 15.50 270.1327.00 21.16 311.0027.50 24.68 323.0328.47 31.63 355.4829.10 38.30 385.4132.50 — 80.97 560.00Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 56The two computed A - curves (points connected by straight line segments) areshown in Figure 4.5. One of them assumes that all odd harmonic components of the currentare present in the measured values, and is therefore incorrect. The other curve is the correctone; it has been produced with the algorithm of Section>0)o- 80.0C-j60. 40.0 60.0 80.0 100.0 120.Cu rrent(A)Figure 4.5: - i1z curve.It can be seen that the correct curve goes deeper into saturation. For the highest fluxvalue, there is a difference of approximately 14% between the peak currents of the incorrectcurve and the correct one. The piecewise linear v - Ir1 curve is shown in Figure 4.6.-—9—- “incorrect”“correct”Chapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 5750.0040.0030.003 20.00>10.002.0 4.0 6.0 8.0 10.Cu rrent(A)Figure 4.6:V—ir1. curve.In the iteration scheme of Section 4.3.2, harmonics up to the 99th order wereincluded, which is more than needed in practice. An average of 23.86 iteration steps wasnecessary (the maximum number of iterations was 34). In order to check the numericalaccuracy of the method, the mis line currents were recomputed back from v - i and A -curves. Numerical errors were found to be very small (less than 0. 001%).4.5 SummaryAn approach to the computation of instantaneous saturation curves of delta-connected transformers has been presented. It uses positive sequence excitation test data asinput, and is suitable for situations in which the tests are performed with a closed delta.For the case study presented in Section 4.4, it was shown that a difference ofapproximately 14% between the peak currents of the incorrect curve and the correct oneoccurred. The last flux linkage point is around 1.2 p.u. In fact, for transient studies, it isChapter 4. Saturation Curves ofDelta-Connected Transformers from Measurements 58often necessary to know peak flux-peak current curves beyond that point. The usual way isto extend the curve up to a value necessary for the study (this extension is sometimes donewith a straight line passing through the previous to the last and the last point in the peakflux-peak current curve). This may lead to larger errors for the peak values of the current.The curves diverge as the flux goes up towards deep saturation. The transformermagnetizing current would always be underestimated if triplen harmonics inside the deltawindings were not taken into account. Errors can also affect the air core reactance value. Aparametric study was done considering typical air core reactances from 0.2 p.u. to 0.5 p.u.,connected to the last point of the correct curve of Figure 4.5. Errors on the slope of thesaturation characteristics, for this case, are between 18% and 25%, when the magnetizingcurve reaches the transformer rated current. These differences may be important inferroresonance or inrush current studiesFor the development of the algorithm in this chapter, it was assumed that each phasebehaves independently. This is valid when the transformer is saturated only. The current inthe saturated phase is much larger than the current in the remaining phases. The saturatedphase can be considered “decoupled” from the other phases.Chapter 5Hysteresis andEddy Current Losses in Iron Core5.1 IntroductionIn this chapter, a general discussion of hysteresis and eddy losses in iron core ispresented. RL networks, in which the inductances are nonlinear and the resistances arelinear, are developed to model the nonlinear and frequency-dependent effects of thetransfonner core (saturation, eddy currents and hysteresis). It is assumed that the core loss isknown as a fhnction of frequency. Simulations are compared to laboratory measurement ofinrush current and to a ferroresonance field test.5.2 Frequency-Dependent CoreModelsRosales and Alvarado[l 8] represented eddy current effects in the core by solvingMaxwell’s equations within the laminations assuming that the permeability and conductivityof the material were constant. They derived expressions for the lamination impedance Zi(jo)and admittance Yi(jo) in the forms:Z1(jw)= 1tanh(), (5.1)* 5cChapter 5. Hysteresis and Eddy Current Losses in Iron Core 60andY,(jo)—= (1/R)eoth() (5.2)whereR,=- =d%Jjo4wand1= lamination length;w = lamination width;2d = lamination thickness;= magnetic permeability of the material;a = electric conductivity of the material.From the expansion of the hyperbolic tangent in (5.1) into partial fractions1tanh=2 2’ (5.3)kI2 + [r (2k — 1)/21they realized a series Foster-like linear circuit and from the expansion ofthe hyperboliccotangent, they realized the parallel Foster-like circuit of Figure 5.1. The accuracy of therepresentation depends on the number of terms retained in the partial fraction expansion.Chapter 5. Hysteresis and Eddy Current Losses in fron Core 613+5+7+ ,9+R1LFigure 5.1: Eddy Current representation of the core after [18]Tarasiewicz et al.[17] used (5.1) and expanded the hyperbolic tangent in continued fractionform,tanh= , (5.4)1+and realized the ladder network ofFigure 5.2, known as the standard Cauer circuit.R1 R2 R3Figure 5.2: Eddy Current representation of the core after [17]They observed that the number of terms to be retained in the continued fraction of (5.4) issmaller than the number of terms to be retained in the partial fraction of (5.3) for the samefrequency range and verified that the four sections of the continued fraction model of FigureChapter 5. Hysteresis and Eddy Current Losses in Iron Core 625.2 would be enough to reproduce the exact impedance of (5.1) for a frequency range up to200kHz with an error less than 5%. They also noticed that to achieve the same accuracy, 72sections of the Foster-like circuit of Figure 5.1 would be required. Based on that, theydecided to work with the standard Cauer circuit. However, it seems the authors were notaware that once the circuit parameters of Figure 5.2 are known, a Foster-like equivalentcircuit which reproduces the same frequency response with exactly the same number ofelements can be realized as well (later on in this chapter, network realizations will beaddressed)[61]. Therefore, it seems there is no apparent gain in choosing this ladder networkto represent eddy current effects in the core.Dc Leon and Semlyen[20] suggested the Cauer circuit of Figure 5.3. The RLparameters were obtained by a nonlinear fitting process to match (5.1). This circuit can beinterpreted as a discretization of the lamination. The inductances represents the flux pathsand the resistances produce the eddy loss. At DC excitation, the current flows through themagnetizing inductances lying longitudinally (in the previous circuits, the DC current flowsthrough one inductance only). This could be interpreted as a uniform flux distribution in theentire lamination at DC level.Figure 5.3 : Eddy Current representation of the core after [20]5.3 Core LossIn the algorithm developed in chapters 3 and 4, the excitation branches oftransformers were represented by a nonlinear resistance in parallel with a nonlinearL1 LChapter 5. Hysteresis and Eddy Current Losses in Iron Core 63inductance. Thus, hysteresis and eddy current loss were assumed to have the samefrequency-dependence. It is clear that the circuit would not reproduce the core loss forfrequencies different from the rated frequency.A more detailed core model should take into account the nonlinear and frequency-dependent effects of saturation, hysteresis and eddy currents. It should be emphasized thatinformation about how eddy and hysteresis losses are split, is in fact artificial. Today, thepoint of view of many researchers is that virtually all the observed losses are resistive lossesassociated with micro eddy currents due to Barkhausen jumps of domain walls [64,65], nomatter how slow the magnetization ioops are traversed.Core models in which hysteresis and eddy currents are treated simultaneously aredeveloped in the next sections. A Foster-like circuit is used to represent both, hysteresis andeddy current effects. First, consideration is given to the core operating in the linear region.Nonlinear effects are considered later.5.4 Eddy Current andHysteresis ModellingThere is a property of linear passive networks in which the knowledge of the realpart of an impedance Z(s) for sjw (Re (Z(jco)), completely determines Z(s) [66]. If (Re(Z(jo)) is a given rational function of frequency w, then one can construct the correspondingrational function Z(s) in terms of the complex frequency variable s. Once Z(s) is known, itcan be realized as a passive network [61,63,66]. The same process applies to the admittancefunction Y(s), as described next.Chapter 5. Hysteresis and Eddy Eurrent Losses in Iron Core 645.4.1 Construction of Y(s) from Its Real PartThe process of constructing Y(s) or Z(s) from their real parts is well known innetwork synthesis theory and only a brief discussion is presented here. For more details, thereader is referred to [61,66].Consider first a simple circuit composed of only two linear RL elements connectedin parallel (Figure 5.4a).(a)AFigure 5.4: Core parameters.Its admittance Y(s) could be written in the formIt follows thatY(s) = +(b)IG1 Y(s) + Y(—s)R 2If the elements are frequency-dependent, G(co)= 1/ R(o) Re[YU)], or with sjw,Re{Y(s)]= Y(s) +Y(—s)(5.5)1(5.6)Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 65Now, consider the exciting branch of a single-phase transformer to be linear (theextension of the method to include saturation effects will be done later). One can visualizethe core operating below the knee of the saturation curve. In this region, the saturationcurve could be approximated by a linear segment (dashed line of Figure 5.4b). The excitingbranch can be represented by a frequency-dependent lossy element R(co) in parallel with themagnetizing inductance L(w). The relation which ties R(co) to L(o) is shown below.First represent Y(s) as a rational function containing n pairs of poles and zeroes:— N(s)— K(s + z1 )(s +z2). (s + Zn)5 7(s)— D(s) — (s+p1)(s+p2)• (s + pj’)i and z are real and positive, i.e., the poles and zeroes ofY(s) lie on the left hand side of thes plane. From (5.6) and (5.7), the real part ofY(s) is expressed by:1 E N(s)D(—s) + N(—s)D(s)Re[Y(s)]= D(s)D(—s)or— U(S2)— K(_s2 + u)(_s2 + u)• . . (—s + u)Re{Y(s)]— W(s2) — (—s2 + p)(—s2+p)• . (—s2 +p)‘ (5.8)where u is also positive and real [621. Expanding (5.8) into partial fractionsK K KRe[Y(s)]=K0,,+ (s22)+2 •+ ( 2)(5.9)Further expanding each fraction of the equation above into two partial fractionsRe[Y(s)] = K, ++ K1 / • + Kn + K,, I 2p,,(s+p1) (—s+p1) (s + P) (—s + p,,)Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 66or(K K /2p K 12p (K K1 /2p K /2pRe[Y(s)]=I—-+ I ?‘ +.“+ “ (5.10)j2 (s+p1) (s+p)} 2 (—s+p1)’ (—s+pj)Comparing (5.6) and (5.10) it is clear that the admittance function is given by:(5.11)(s+p1) (s+p2) (s+p)From (5.11), one can also write the admittance Y(s) in the form:a s” i-a + a s+a22 (5.12)bs”+•. +Is+b0It can be seen that if the function G(w)=Re{YC,)] is known, Y(s) is determined.’ Itshould be pointed out that G(w) must be expressed as an even rational function in w2 withreal coefficients in the form of (5.8) with w2= -s2 and 0G(co)<oo for all frequencies [66]. Acurve fitting procedure to obtain G(o) or R(o)) in the required form, from one of the coreloss - frequency curves supplied by steel lamination manufacturers, is explained in AppendixD. It is recommended that the data be taken for an induction level, at which the lamination isnot saturated.5.4.2 Linear Network SynthesisFrom Y(s), any of the RL equivalent networks of Figure 5.5 can be realized with thesame minimum number of elements. If for any of these circuit, the parameters are known,Y(s) or Z(s)=1/Y(s) could be determined and the remaining circuits could be realized as1 The construction of Y(s) from its real part in the partial fraction form of (5.11) was first suggested byBode[58]. There is a method to compute the coefficients a and b in (5.12) directly from (5.8) due to C.M.Gerwetz [57]Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 671 1/LY(s) = -— +rç0 1=1 s+R/L,R L‘K1R R R_____Ia) Parallel RL Foster-like networkL L1c) Physical Cauer networkandR1 R1L1 L1b) Series RL Foster-like network.d) Standard Cauer networkwell. It must be clear that all these circuits are equivalents only with respect to theirterminals, i.e., the network elements of each circuit must have the prOper values to producethe same terminal impedance or admittance as a function of the con4lex frequency s.Let us describe the algebraic process for realizing the parallel Foster-like circuit. Theterminal admittance of the RL network ofFigure 5.5 a isThis circuit could be realized directly from (5.11) with(5.13)RFigure 5.5 : Realization ofRL networks.Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 68For the Cauer circuit ofFigure 5. 5c Y(s) is written in the continued fraction form:1 11 (5.14)Ls+1 1R L_1s+11RIs+R1Equation (5.14) could be derived from (5.12) according to the following steps:a - from (5.12) Y(s) is split into two terms by removing the constant a,/b;a a s’’ +• • + a’s2 + a’s + a’Y(s)=--+“s+•••+b21b0°(5.15)or2b bs+•”+b2+bis+b0a1s’+ +as2+as+ab - removing4—s from the denominator of the second term of the equation above, resultsa_111 . (5.16)a1 a,1s’+• ‘ ‘+as2 +as+ab’ s’+•’•+b’s2+b s b2 1Now, the process of long division is done for the term between the brackets ( steps a and bare repeated). For each cycle, a constant (step a) and a pole at s=oo(step b) are removed.The process continues until the order of the polynomial in the last continued fraction is one.For the remaining circuit realizations ofFigure 5.5, the reader is referred to [61].Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 69A remark must be made about the calculation of Y(s). Consider the Foster-likecircuit of Figure 5. 5a with an inductance Ld placed across its terminals (Figure 5.6). It isclear that Ld does not affect the conductance G(o); it affects only the susceptance. In theprocess described earlier, Y(s) is the minimum admittance function that produces G(o). Thefull admittance is given by Ld in parallel with Y(s).RFigure 5.6: Frequency-dependent representation of the core.5.4.3 Iron Core NonilnearitiesSince the resistances and inductances in the circuit ofFigure 5.6 do not represent anyphysical part of the core, it is not clear how to incorporate nonlinear effects. Consideringthat low frequency elements contribute more to saturation than high frequency elements,only the inductance Ld is made nonlinear. The Foster-like circuit was chosen because it hasonly two nodes, therefore, it is computationally more efficient than the remaining circuits ofFigure Numerical Example - HysteresisIt is appropriate to illustrate how the model works for an example in which a core isassumed to experience only hysteresis loss (this is an arbitrary case since hysteresis and eddylosses cannot be separated). Initially, suppose hysteresis could be reproduced by a resistanceChapter 5. Hysteresis and Eddy Current Losses in Iron Core 70in parallel with a nonlinear inductance. For a sinusoidal applied voltage, at rated frequency,the flux-current loop ofFigure 5.7 is obtained. Let us keep the flux amplitude constant.2 Forany frequency, the area inside the flux-current loops should be the same and the loss is afunction of how fast the trajectories are traversed. For a frequency equal to twice the rated ifrequency, the voltage doubles. So, hysteresis resistance would have to be twice thehysteresis resistance at rated frequency to produce the same loss per cycle. For anyfrequency, the resistance R should be replaced by an equivalent frequency-dependentresistance R(o.) according to:R(O))Rr, (5.17)where cor is the rated angular frequency and 14 is the hysteresis resistance at ratedfrequency.iLet us now represent the core by a two slope A-i curve, defined by points in Table 5.1, anda resistance Rr=100 2 at rated frequency (60Hz). Using the method described in theprevious sections with Wh>>Wdy (Appendix D), the Foster-like circuit parameters ofFigure2 This could be acomplished by keeping constant the ratio between the amplitude of the voltage signal andthe frequency.Figure 5.7: A—i hysteresis curve.Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 715.6 are found for a frequency range from 60 Hz to 3 kHz. The fitted frequency-dependentresistance and its value from (5.17) are shown in Figure 5.8.Table 5.1: Flux-current curveCurrent (A) Flux (V.s)0.0000 0.000020.0000 0.4000300.0000 0.60005—Exact CurveI 31I ‘IFrequency (kHz)Figure 5.8: Frequency-dependent resistance.Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 720.50.0-0.550 -50 0 50Current (A)b) Frequency-dependent resistanceFigure 5.9: Flux-current loops.The flux-current plots shown in Figure 5.9 were obtained from time-domain simulationsusing Microtran®[431. Sinusoidal flux linkages of constant amplitudes and frequencies of 60Hz, 180 Hz and 300 Hz, were used for the simulations. In Figure 5.9a, the loops wereobtained considering the core loss represented by a constant resistance and, in Figure 5.9b,with the frequency-dependent model developed here. In Figure 5.9b, the area of the loopsare nearly independent of the frequency, however, the flux-current loops tilt slightlyclockwise as frequency increases. In a qualitative way, one could interpret that an increaseof the core resistance is associated with a decrease in the core inductance caused by theflux being pushed away from the center of the lamination.5.5 Inrush Current: Simulation andMeasurementsA single-phase 60 Hz, 1 kVA, 208/104/120 V laboratory transformer, in which thecore is made of non-oriented silicon steel laminations, was tested. From short-circuit tests,with the voltage source connected’to the 104 V side:0.50.0-0.5-50 0Current (A)a) Constant resistanceChapter 5. Hysteresis and Eddy Current Losses in Iron Core 731. AC resistance measured with the 120V side short-circuited - 0.60 2;2. Leakage reactance with the 120 V side short circuited - 0.19 2;3. DC winding resistance - 0.40 2.The points (empty circles) of the flux-current characteristic of Figure 5.10 wereobtained from no-load loss measurements and V,-I values, using the method developedin Chapter 3. When the core is operating in the unsaturated region, its inductance can beroughly approximated by 2.4 H. The core loss at 60 Hz is represented by a constantresistance of 1.8 k 2. All parameters are referred to the 104 V terminal.The Foster-like circuit linear parameters shown in Table 5.2, was calculated for afrequency range from 60 Hz to 3 kHz assuming the ratio of hysteresis to eddy loss per cycleto beW11/ eddy=2 at 60 Hz [67] (it does not make any difference in the simulations if thisratio is taken as 1 or 3). The transformer was initially demagnetized and later, a voltagesource was switched in the 104 V terminal. The voltage signal (Figure 5.11) was saved as anASCII file and used as input in Microtran®. The inrush current waveform was measured aswell. Plots of the simulated current using two core models (Foster-like circuit and a constantresistance in parallel with a nonlinear inductance) and the measured current using a time stepof 50 l.ts, are shown in Figure 5.12. For the study, a straight line segment was connected tothe last point of the curve ofFigure 5.10. Its slope was chosen to match the first peak of themeasured current, when the flux linkage reached its first peak (the flux waveform wasobtained integrating the voltage waveform ofFigure 5.1 1).33 By extending the last segment of Figure 6.10, it was found that the first current peak in the simulationswas underestimated by nearly 5%. The winding resistance, used for the simulations, was 0.45 12 which isslightly above the measured value.Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 74Table 5.2: Linear circuit parametersFoster-like circuit parameters /Ld=2.4000H R1= 1722.4899L1=4.1518H R2=22264.6946 2L2=4.6595H R= 5271.37022In general, the agreement between measurement and simulations, for any of the usedmodels, is good. Figure 5.13 is a magnification of Figure 5.12 for the time interval between0.15 and 0.20s. The two core models produce nearly the same response and are in goodagreement with measurements. Another simulation was performed representing the core by anonlinear inductance only. The current waveform was nearly the same as the constantresistance curve ofFigure 5.13.Current (A)Figure 5.10: Flux-current curve of a single-phase 1 kVA transformer. 10 15 20Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 75160 —120 —80 —40—0—-40 —-80 —-120 ——160—I I I0.05 0.10 0.15 0.20 0.2Time(s)Figure 5.11: Transformer input voltage.140-120 —Constant Resistance100-Measured Curve80 —Foster-like Circuit60-:::I I I0.05 0.10 0.15 0.20 0.25Time(s)Figure 5.12: Transformer inrush current.Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 7686200.20Figure 5.13: Transformer inrush current.5.6 FerroresonanceThe BPA system of Section 3.4 is simulated once more. Now, the transformerexciting branch is represented by a Foster-like circuit with the parameters calculated for afrequency range from 60 Hz to 3 kHz, assuming the ratio of hysteresis to eddy loss percycle to be Ww’Wdd=l at 60 Hz [70]. The simulation and field test results are shown inFigure 5.14. The Foster-like circuit produces nearly the same response as a linear resistancein parallel with a nonlinear inductance (see Figure 3.8 in Chapter 3).- — — — —- Constant ResistanceMeasured CurveFoster-like Circuit0.15Time(s)Chapter 5. Hysteresis and Eddy Current Losses in Iron Core 77- Field Test1200 - Foster Circuit800-400 -0--400---800-‘ I I I I20 40 60 80 100 120 14Time (ms)Figure 5.14: Ferroresonance in a power system.5.7 SummaryAn approach to model frequency-dependent effects in the transformer core from coreloss data was presented. The parallel Foster-like circuit model, in which hysteresis and eddycurrent effects are treated simultaneously, was realized. Theoretically, the circuit models canbe used for any frequency range (minimum frequency must not be zero).If the circuit model is used to represent a core in which the hysteresis loss isdominant (e.g. amorphous core [221), the core loss per cycle is nearly independent of theChapter 5. Hysteresis and Eddy Current Losses in Iron Core 78frequency. The flux-current trajectories are generated by the circuit models with no need topre-define them.An inrush current case and a ferroresonance case were used as test examples. Forboth cases, there were no major improvements in representing frequency-dependent effects.in the core. For the inrush current case, the simple model in the form of a nonlinearinductance reproduced the measured current reasonably well. For the ferroresonance case, anonlinear resistance (Chapter 3) in parallel with a nonlinear inductance represented the corereasonably well.It is still premature to say that frequency-dependent effects in the core are notimportant for all transient cases. More validation tests against other field measurements needto be made in the fhture, to filly determine their importance for practical applications.Chapter 6 /TransformerModels - Applications6.1 IntroductionTransformer models based on parameters obtained at 60 Hz, and suitable forfrequencies up to 5 kHz, are presented. Frequency-dependent effects in the windings andcore nonlinearities are discussed. These models were developed for harmonic studies, as partof a research project for The Canadian Electrical Association, but may be used for transientstudies as well. To validate the models, many tests were carried out on several transformerunits of different power ratings. Sensitivity studies are performed to show how the totalharmonic distortion of current and voltage behaves as the transformer parameters change. Afield test involving a transformer, as part of a power system feeding a DC drive, is describedas well, for which simulations agree reasonably well with measurements.6.2 Basic Transformer Equivalent CircuitFigure 6.1 is the basic transformer model. Z(jo) represents the frequency-dependentshort-circuit impedance. The excitation branch is represented by nonlinear RL elements obtainedfron measured rms values of voltages and currents(V,mf (Lmj) and no-load losses with the79Chapter 6. Transformer Models - Applications 80conversion technique of Chapter 3. These elements should be placed at a point in the equivalentcircuit where the integrated voltage is equal to the iron core flux. This point depends on thetransformer design and is usually not accessible in the model. Howev’er, it can be approximatedfairly accurately by connecting the elements to the winding closest to the core (usually the lowest /voltage winding) [4].6.2.1 Single-Phase Two-Winding TransformerConsider initially a single pair of magnetically coupled coils as shown in Figure 6.2a.This network can theoretically be described in terms of self and mutual impedancesrv1l Lz, z12Ti1[‘j[z2, z2214j => [v]—[z][i}, (6.1)or with its inverse relationshipri1i EI 1=1iI = [i]=[y][v], (6.2)L 21 L’21 221’2lwhere Z12 = ZM and [Y][Z]1. V2’ and 12’ are the secondary voltage and thesecondary current referred to the primary side, respectively. The elements of [ZJ can befound from open-circuit excitation tests. For instance,v2’Z21=y12—0Figure 6.1: Basic transformer model.(6.3)Chapter 6. Transformer Models - Applications 8112’ 1ZSH12..........., 4v11 1v2 vu____1v21(a) (b)Figure 6.2: Two winding transformer:a) Pair ofmagnetically coupled coils;b) Equivalent circuit referred to primary.Unfortunately, in practice the magnetic coupling is so high that Z21 is approximatelyequal to Zn. [Zj is then almost singular and its inversion becomes ill-conditioned (one shouldnote that Z11 Z22). The short-circuit impedances, which are more important than themagnetizing impedances in most studies, get lost. One suitable way to overcome thesedifficulties is to use the branch admittance matrix [Yj of equation (6.2). Although [Z]becomes infinite for zero exciting current, the [Y] matrix exists and can be found fromstandard short-circuit tests, as shown later.For a single pair of magnetically coupled coils, only one short-circuit impedance ZSHexists (Figure 2b). Then, the short-circuit admittance YSH can be obtained from(6.4)To construct [F], the four elements )‘ii, l’12, l13, and Y14, are needed. They can beobtained directly from Figure 6.2b.’ The 2x2 [F] matrix is then found from one short-circuitadmittance element as1 It is useful to note that Y11 is the sum of the admittance element connected between the two nodes (YsH)plus the aclniittance element connected between node 1 and the reference node (since we are neglecting theexciting current, its value is nil). We can also see thatY1Y22.Chapter 6. Transformer Models - Applications 82ri1i EH -11T‘SH 1’J (6.5)ZSH is frequency-dependent and can be reproduced reasonably well by the two branchparallel Foster-like circuit of Figure 6.3a or the series Foster-like circuit of Figure 6.3b asrecommended by CIGRE[54]. R5 and L are the resistance and leakage inductance of thewindings at rated frequency. R in parallel with the 60 Hz leakage inductance produces thefrequency-dependent effects.(b)Figure 6.3: Frequency-dependent short-circuit impedance model.6.2.2 Single-Phase Three-Winding TransformerSingle-phase distribution transformers usually have a grounded tap at the center ofthe secondary terminals (Figure 6.4a). These transformers are modelled as three-windingtransformers (Figure 6.4b). The impedances Z8, Zpr and can be found from three short-circuit tests. Unfortunately, these test data are usually not available[4]. The only test usuallyavailable is the impedance measurement carried out with the secondary and tertiary terminalsgrounded to the center terminal. Fortunately, it appears that an accurate value of the shortcircuit impedance between primary and secondary is not needed for practical applicationsSince the current flows mainly through the impedances Zps and ZPT (see Figure 6.4b). It isRII ‘2(a)Chapter 6. Transformer Models - Applications 83therefore reasonable to assume Zps ZPT Zsr referred to the same terminal. In order toproduce the frequency-dependent effects in each winding, each branch Z3, ZPT and ZST isreplaced by the equivalent circuit of Figure 6.3. The core nonline’ar elements are obtainedfrom typical V, - I, curves, as explained in Chapter 3.P Ps SI(a) (b)Figure 6.4: Three-winding transformer.a) Coupled coils;b) Equivalent circuit.6.2.3 Three-Phase TransformerThree-phase transformers are best represented in matrix form. It is simple to extendthe single-phase transformer formulation to three-phase two-winding transformers. Equation(6.2) becomes a matrix equation[YSH]=[ZSH]’. (6.6)For a short-circuit test performed on a two-winding transformer, the voltages andcurrents in each phase of the feeding terminal are related according to[VABC} = [ZSH][IABC]TorJ,4I=I=O;VBZBA=j-‘B’C°vcChapter 6. Transformer Models - Applications 84VA ZM ZAB zAc1 ‘A= ZBA ZBB ZBC ‘B (6.7)V Z Zc Zj 1c /To obtain each element of [Zszqj in equation (6.7), the short-circuit tests are doneaccording to the circuit of Figure 6.5.‘AVAThree-phaseTransformerFigure 6.5: Three-phase transformer short-circuit test. -A voltage signal is applied to one phase at a time and the current in that phase andthe voltages in A, B and C are then measured. The first column of [ZSH] can be obtained by:andThe same procedure applies to the other two columns. Due to symmetry, only sixmeasurements are required to construct [Zsn-j. In practice, [ZSH] is generally assumed to bebalanced (Z ZAC =ZBC and ZM=ZBB=ZCC) and the number of measurements is reduced totwo. Positive and zero sequence short-circuit tests are the standard data required to build[ZSH] 22 transformers rated 500 kVA and below, only positive sequence tests are available[551.Chapter 6. Transfonner Models - Applications 85As explained earlier, since the impedance matrix may become singular, it isconvenient to use admittance matrix [Y] formulation. In fact, here we use matrices [R8j and[Lp f’ or [coLp f’ (imaginary part of [Y]). These matrices are obtained from positive andzero sequence tests and computed by the support routine BCTRAN [4]. [Rs] is a diagonalmatrix obtained from positive sequence tests - the zero sequence resistance R is assumedto be the same as the positive sequence resistanceR5÷. The manufacturer provides either theR+ in ohms or the load losses from which R5+ can be obtained. Once the matrices aredetermined, arbritary transformer connections can be realized by node renaming.Frequency-dependent effects in the short-circuit impedance are treated the same wayas for single-phase transformers. Here the elements of Figure 6.3a are all matrices. [Rn] iscalculated according to the equation[Re] = k [oL], (6.8)Where k is a constant and [oL] is the reactance matrix obtained from positive and zerosequence short-circuit reactances. For the representation of the circuits in admittance form,the conductance matrix [G] can be used instead of [Re], where,[G] = [oiLJ’The proper way to represent saturation effects is to look at the transformer magneticcircuit. Saturation is related to fluxes in the core and tank. There are several types of coreconstruction. To model the core, one should know what kind of geometry the transformerhas. One should know if the transformer is constructed with three different cores, if its coreis three-legged, five legged or shell-type. A zero sequence magnetizing impedance can beestimated for each type of core construction.3 It can be approximated as linear, because of3This probably is not applicable to five-legged transfonners constructed from four different cores [69].Chapter 6. Transformer Models - Applications 86fluxes usually passing through air between core elements and the tank. The saturation curvemust be obtained for each leg of the transformer and placed across each winding at thesecondary terminal to represent the nonlinear effects[4]. If detail of core design are notknown, the positive sequence saturation curve is placed across each winding.6.3 Estimation ofTransformer ParametersMeasurements were taken to determine the values of the parameters in thetransformer equivalent circuit, and to correlate them with the nameplate data. Single andthree-phase transformers were tested at Powertech Labs and BC Hydro and PowerAuthority (Table 6.1). Short-circuit impedance measurements and open-circuit saturationtests (V,.,,,=f (I,,,,.) and no-load losses), were performed for the units listed below. The basesof the per unit (p.u.) system used for the short-circuit impedances are rated voltage andpower.6.3.1 Short-Circuit TestsHigh frequency currents produced by the signal analyzer HP3562A4 were injectedinto the transformers under test. Short-circuit frequency response impedance measurements(real and imaginary values) were recorded as ASCII files on floppy disks. Single-Phase TransformersIn Figure 6.6, some of the measurements of impedances made on the primary side, withthe secondary side short-circuited and the tertiary left open, are shown. R(J) and X(/) represent4 analyzer has a built-in signal source which can produce and measure a scanning signal well above thefrequency range of interest. Its measurement range is 6411 Hz to 100 kHz.Chapter 6. Transformer Models - Applications 87the real and imaginary parts of the short-circuit impedance, respectively. Although thetransformers have the same rated power and nearly the same short-circuit impedances at 60 Hz,there is a considerable difference in the frequency response behaviour df transformer 1 comparedto the others. Differences in coil and core constructions may be the causes of these discrepancies.Impedances measured on the secondary side, with the tertiary short-circuited and anopen primary, are shown in Figure 6.7. It is difficult to correlate Figure 6.6 and 6.7. X(/) inFigure 6.7 is, for the whole frequency range, slightly bigger than X(/) in Figure 6.6(transformer 1 behaves in the opposite way). Comparing the measurements of Figure 6.6and 6.7, it can be seen that Zps is almost the same as ZST for transformers 2 and 3. Forpractical purpose, it is reasonable to assume ZPSZST.Table 6.1: Distribution transformersManufacturer (date) Power Voltage ZSH* Measured 60 Hz(kVA) (kV) (%) ImpedanceCode ZSh (%) R (%) X (%)xformerl 1975 50 14.4 1 2.18 - - 1 -xformer2 May 1992 50 14.4 1 2.10 - - 1 -xformer3 1978 50 14.4 2.10 -- 1 -xformer4 1973 50 7.2 2.30 - - -xformer5 not available 75 14.4 2.83 2.78 1.18 2.52xformer6 not available 100 14.4 2.10 2.03 f__0.77 1.88xformer7 not available 167 14.4 2.10 [ 2.26 0.68 2.16xformer8 not available 10 14.4 2.20 1.91 1.10 1.56xformer9 not available 25 14.4 2.10 2.01 0.89 1.80xformerlo not available f 50 14.4 2.10 1.90 0.81 1.72xformerll f May1992 10 14.4 2.20 - - - Ixformerl2 ** not available 150 7.2 4.21 -- [ - Ixformerl3** not available 500 7.2_[ 5.60- - I[jtrmer14** not available 75 [ — 14.4 [ - -_[ ZZ[_-_I*nameplate short-circuit impedance * *three_phase transformers541Chapter 6.xfoi±er 1:::::::J::::::::::..xftdner.3Oliner4Transformer Models - Applications 8815xfornier 1xformer2L xfornier3/4 5I1 23 4 5 1 2 3Frequency (kHz) Frequency (kHz)Figure 6.6 : Impedance measured on the primary sidevfnrnwr1zz r2 ,L..bcormer3------1---. bcomier4 ‘I’ :ii /_•/-:___±iz____t_____15105xförm———.— i’dformer2..:::::::.. .::-.-.-- dfonner4//‘I,, ..,.,V.-.—,—_.--i.-5410.50.40.3020.151 2 3 4 5 1 2 3 4Frequency (kHz) Frequency (kHz)Figure 6.7: Impedance measured on the secondary side.2.01.5‘ 1.00.52 3 4 5 1 2 3 4 5Frequency (kHz) Frequency (kHz)Figure 6.8: Short-circuit impedances of single-phase transformers.Chapter 6. Transformer Models - Applications 89In Figure 6.8 one can see the real and imaginary parts of the short-circuit impedances(transformers 5, 6 and 7) measured the way in which manufacturers probably supply the data tousers (measurements are taken on the piimary side with the secondary’ and tertiary grounded tothe center terminal). The equivalent impedance seen by the source is Zp in parallel with Z1 inthis case. Measured and simulated L/R curves are shown in Figure 6.9. The solid curves areobtained from measurements. The simulated curves are produced by two parallel circuits of thetype ofFigure 6.3a.:::::::::::::::::::EE:::.:::.:R = Rp = 13 p.u.. (6.9)R (R I 2)(coL)6 10R(f)- 2 + R+(oL)21.0 1.0 1.0Frequency (kHz) Frequency (kHz) Frequency (kHz)Figure 6.9: Measured and simulated LIR curves.Two values of resistances Rp are used [54]:R = Rpm = 30p.u. andThe real and imaginary parts of the computed short-circuit impedance are calculatedas follows:xformer5 (75 kVA) xformer6 (100 kVA)0.0100.001xformer7 (167 kVA)‘-. \Meaure11çpMeasured’RpmaxRpniin -andChapter 6. Transformer Models - Applications 90(R2 / 2)(coL )— R+(wL)The 60 Hz resistance R is obtained from load losses. ‘It can be seen that thesimulated L/R ratios come close to the measured L/R curves (especially for R Rpmin). Fortransformers 8,9 and 10 (Figure 6.10) one can see that the X(/) curves are bent upwards,indicating a resonance frequency near the last measured frequency. Stray capacitances needto be included in the model to produce the correct frequency response when one gets closeto resonance points. The first resonance frequency for all tested single-phase transformers isbetween 6 and 60 kH.z. A similar resonance frequency range was found previously byOntario Hydro[56].0.5 50.4 11 2 3 4 5 1 2 3 4Frequency (kHz) Frequency (kHz)Figure 6.10: Short-circuit impedance frequency response. Three-Phase TransformersThe short-circuit impedance measurements were performed according to the circuitof Figure 5.5. The impedance measured at each phase (self impedance) of transformers 12,13 and 14 are shown in Figure 6.11, 6.12 and 6.13. The mutual impedances were measuredas well and found to be negligible for the 500 kVA and 150 kVA transformer (they were lessthan an order ofmagnitude of the self impedances).5Chapter 6. Transformer Models - Applications 91302010— RaaRbbRcc-XaaXbbIx20100-10S74—-201 2 3 4 5 1 2 3 4Frequency (kHz) Frequency (kHz)Figure 6.11: 150 kVA three-phase transformer (self impedances).1.55Raa-- RbRcc1.00.5Xaa- -- ... XbbXcc651 /Z/// 4 5 1 2 3 4 5Frequency (kHz) Frequency (kHz)Figure 6.12: 500 kVA three-phase transformer (self impedances).0.6 .I‘s00 ---——-0.4-0.6::Y1 2 3 4 5 1 2 3 4 5Frequency (kHz) Frequency (kHz)Figure 6.13: 75 kVA transformer short-circuit impedances( self impedances).Chapter 6. Transformer Models - Applications 920.0100.001Frequency (kllz)Figure 6.14: 500 kVA transformer LIR curve.Each phase of the 500 kVA or 150 kVA transformer could therefore be simulated asa single-phase two-winding transformer. In Figure 6.14, simulated and measured (solid lines)L/R curves for the 500 kVA transformer are shown. R is obtained from equation (6.9).Simulated curves come close to the one obtained from measurements (especially for R=Rpmax). For this example the parameter k in equation (6.8) is:30k=L,where wL+ is the positive-sequence short circuit reactance at rated frequency.For the 75 kVA transformer, the mutual impedances were of the same order ofmagnitude of the self impedances.The matrix form would therefore be more appropriate torepresent this transformer. Unfortunately, for three-phase distribution transformers, onlypositive sequence data are usually available from manufacturers.6.3.2 Open Circuit TestsIn the open circuit tests, the low voltage winding is excited with a nearly sinusoidalvoltage source at 60 Hz to obtain the transformer saturation characteristics. No-load losses1.0Chapter 6. Transformer Models - Applications 93and mis current were measured for different rins voltage levels. The x-axis of Figure 6.1 5ais the ratio of the rms exciting current I to the mis exciting current ‘crafed at ratedvoltage for single-phase transformers 1,2,3,4 and 8. The y-axis ‘is the ratio of the rrnsexciting voltage V to the rated voltage Vrad. The solid line is the average curve. Belowrated voltage, saturation curves differ very much from each other. However, for harmonicand transient studies, the representation above rated voltage is more important and in thisregion the curves come close to each other.P.(a) (b)Figure 6.15: Single-phase transformers.a)V, -I characteristics;b) V, - average power curves.The x-axis of Figure 6.1 5b is the ratio of the no-load exciting losses P to the no-load losses at rated voltage Vrated for the same transformers. The solid line represents theaverage curve.For three-phase transformers open-circuit tests are usually performed for positivesequence. Figure 6.16 is the measured positive sequence excitation characteristic of thethree-phase 75 kVA transformer. one can see how the exciting current through each phase(solid line is the average curve) behaves as the exciting voltage changes.0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5Iexc/Iexc-rated Pexc/Pexc-ratedChapter 6. Transformer Models - Applications 942 4 6 8 10Iexc/Iexc-ratedFigure 6.16: V, -I characteristic for a 75 kVA three-phase transformer.Typical distribution transformer correction factors that should be applied to no-loadlosses at rated voltage to give the correct losses when the transformers are driven intosaturation, are shown in Table 6.2[441. Agreement between this table and the solid linecurves ofFigures 6.15 and 6.16 is reasonably good.Table 6.2 -Correction factorsCorrection FactorsOperating Voltage (°“) No-load Exciting kVA ratingLoss Current105 1.15 1.50 For 167 kVA and below (1)110 1.30 2.20 For 150 kVA and below (30)105 1.50 2.30 For 250 kVA and above(10)110 2.40 4.60 For 225 kVA andabove (3k) Stray CapacitancesThe curve shown in Figure 6.17 was obtained from a recently manufacturedtransformer (xformerl 1 - Table 6.1). The rms current at 90% of the rated voltage is smaller thanChapter 6. Transformer Models - Applications 95the rms current at 50% ofthe applied voltage. Here, the exciting current is very much affected bystray capacitances.A crude estimation of the saturation curves and the open-circuit capacitance can befound iteratively. First, the algorithm generates the peak flux-current characteristic“corrupted” by stray capacitance effects as shown in Figure 6. 17b. The peak current isdecreasing in the region between points A and B. Since The inductance is very high in thisregion. It can be assumed that it is infinite between turning points A and B. The peak currentI through the open circuit capacitanceC0 is then the horizontal distance between turningpoint A and any point in between A and B. For instance at point B, the open-circuitedcapacitance can be obtained from:>0)00>EI-I—,______“—‘open— BIn Table 6.3, the “corrupted” flux-current points were obtained using Figure 6.17a asinput data.150100502 4rms Current (A)(a)Figure 6.17:1(b)Newly manufactured transformer.a) V - I,,, curve;b) Corrupted flux-current curve (not to scale).Chapter 6. Transformer Models - Applications 96Table 6.3: Corrupted flux-current curve.Flux Linkage(Vs) Current(A)0.000000 0.00000000.040327 0.42628 10.059046 0.7476570.075777 0.9053 180.095659 1.1933160.114790 1.2407260.151553 0.8153920. 169560 0.3968830.189066 0.1238510.207823 0. 162665The following information is needed:peak current at turning point A = 1.240726 A;peak current at turning point B = 0.12385 1 A;peak flux at turning point B = 0.189066 Vs.Then, the open circuit capacitance referred to the 120 V side is1.240726 — 0.123851Copen = (2irx 60)2 xO.1 89066= 41.56pF.If Copen is needed to be referred to the high voltage terminal,C0,, =41.56x(120/14400)22.89rF.The Copen could then be split into three parts and placed between each transformerterminal and ground of Figure 6.4b. Although the open circuit capacitances affect theexciting current, the first resonance frequency for this transformer was above 6 kHz.Chapter 6. Transformer Models - Applications 976.4 Sensitivity StudyThe frequency-dependent short circuit impedances of transformer 5 are obtainedfrom nameplate data in the way explained in Section 6.3.1. The curve and no-loadlosses are measured in the laboratory and the nonlinear core parameters are calculated.Numerical simulations using the IVIHLF program[57] are carried out to see how the totalharmonic distortion(THD) of the primary current and secondary voltage are affected bychanges in the short-circuit parameter R (see equation 6.11), and in a balanced linear loadconnected half to the secondary and half to the tertiary terminal.5 Stray capacitances areneglected. The transformer primary terminal is excited with a sinusoidal voltage and driveninto saturation. The simulation results are shown in Table 6.4. Rp has little effect in theprimary current and in the secondary voltage. From the last two rows one should note thatat flill load the nonlinear shunt elements are of minor importance (the harmonics producedby them are of small amplitude). The load model is more important than the nonlinearexcitation branch model.Table 6.4 - Sensitivity study.Load Power TFID TNDVoltage Load Factor Primary Secondary(p.u.) (p.u.) Rp(p.u.) Current (%) Voltage (%)1.05 0.00 - 13.0 19.317 0.3121.05 0.00 - 30.0 18.900 0.3171.10 0.00 - 13.0 26.041 0.5421.10 0.00 - 30.0 25.488 0.5501.05 1.00 0.60 13.0 0.041 0.1711.10 1.00 0.60 30.0 0.065 0.280TFI) is the ratio between the total rins value of the harmonics and the mis value of the fundamental [71].Chapter 6. Transformer Models - Applications 986.5 Field TestIn Figure 6.18, three three-phase 500kVA transformers, connected to a 12 kV three-phase power system, feed a 1500 HP DC Drive. Currents and ‘(‘oltage waveforms weremeasured at both transformer terminals. The current and voltage waveforms at the 12 kVside are shown in Figure 6.19 and in Figure 6.20. In Figure 6.21 one can see plots ofmeasurements and simulations of the line voltage at delta side when the DC drive is on lowload. Frequency-dependent effects in the windings and the nonlinear effects in the core areneglected (the transformer is represented as a constant RL short-circuit branch).6 Solid linesare computed curves and dotted curves were obtained from measurements. One can see thatthere is a good agreement between measurements and simulations. The difference betweencurves may be attributed to measurement noise.Figure 6.18: DC drive set up.6Here, the measured priniaiy voltage, current and the short circuit impedance were the required data. Infact, voltage and current waveforms were saved as ASCII files and converted from time-domain to phase-domain to be used as input sources in the MHLF program.12 bV .k500 kVA 600 VP18R._ PDCPCAI4otatT.qveRPWodChapter 6. Transformer Models - Applications 990.01 0.02Time(s)Figure 6.19: Measured voltage waveform.20I .JEfL____0.01 0.02Time (s)Figure 6.20: Measured current waveform.Vab Vca Vbc0.01 0.0Time (s)Figure 6.21: Transformer secondary line voltage.Chapter 6. Transformer Models - Applications 1006.6 SummaryTransformer models for harmonic studies were presented. They are derived from 60 Hzmanufacture?s data. Short-circuit impedance and open circuit saturation tests were performed,the parameters of the transformer equivalent circuits were determined and correlated tonameplate data. The transformer equivalent circuits reproduce the measured short-circuitimpedances reasonably well if stray capacitances are negligible for the frequency range of interestin harmonic studies (60 Hz to 5 kHz). If stray capacitances are important (resonance for short-circuit test close to 5 kHz), it is better to add shunt capacitances to the transformer model. Thesecapacitances are determined by low-frequency or resonance measurements, or taken as typicalvalues from publications or textbooks [58-60].Typical saturation curves for single-phase and three-phase transformers are presented.For situations in which the exciting current includes strong capacitive effects, a crude estimationofthe open circuit capacitance can be made if saturation V,,, - I curves and no-load losses areprovided.Sensitivity studies indicate that the load plays an important role in the harmonic contents ofterminal voltages and currents. In genera], the representation of the core is less important than therepresentation ofthe short-circuit impedance and the load.In general, at steady state, transformers usually do not experience deep saturation(flux in thecore may be only a few percent higher than the rated flux). To predict the exciting current for apractical case is extremely difficult since, even for the same transformer rating (see Figure 6.15), thesaturation curves may differ very much from each other. However, the exciting current, althoughdistorted, is veiy small and will not affect the simulation results. The simplest model of thetransfbrmer was able to reproduce the field test reasonablywell.Chapter 6. Transformer Models - Applications 101It is advisable to include the nonlinear effects ofthe core for situations in which resonance mayoccur ( for instance, the transformer is unloaded or supplying a light load). The transformer could bedriven into a deeper saturated level. The exciting cunent should not be neg’ected here, and similarly tothe inrush current case of the last chapter, in the saturated region, the exciting current is likely to be ‘predicted with reasonable accuracy provided the saturation curves are known up to deep saturation.Chapter 7ConclusionsIron core nonlinearities may play a very important role in transient studies. In thefollowing paragraphs, the major contributions of this research work are outlined.A direct method for producing saturation curves from readily available transformertest data as supplied by manufacturers, was presented. The algorithm is easy to implementand may be useful for electromagnetic transient programs.A method for more correctly representing the saturation curve of a delta-connectedtransformer, suitable for situations in which the tests are performed with a closed delta, wasdeveloped. It uses positive sequence excitation test data as input and takes intoconsideration the removal of triplen harmonics from the line current. The algorithm isiterative and easy to implement.An approach to model frequency-dependent effects in the transformer core fromtransformer no-load loss data, was presented. The parallel Foster-like circuit model, in whichhysteresis and eddy current effects are treated simultaneously, were realized. The fluxcurrent trajectories are generated by the circuit models with no need to pre-define them.102Chapter 7. Conclusions 103Practical applications of transformer models for harmonic studies were presented.The transformer models were derived from manufacturer’s data. and may be used fortransient studies as well. In general, when the transformer is not in deep saturation, the excitingcurrent, although distorted, is very small and will not affect the simulation results. It is advisableto include the nonlinear effects of the core only for situations in which resonance may occur (forinstance, when the transformer is unloaded or supplying a light load).A method for crudely estimating the transformer open circuit capacitance fromsaturation Vrms - I curves and no-load losses, was presented. It is useful for situations inwhich the exciting current includes strong capacitive effects.Two case studies involving transients were investigated: an inrush current test in a 1kVA transformer and a ferroresonace field test. For the inrush current test, the simple coremodel (a nonlinear inductance) reproduced the measured current reasonably well. For theferroresonance test, a simple core model (nonlinear resistance in parallel with a nonlinearinductance) also represented the core reasonably well. Theoreticaly, this core model islimited to a single frequency. At frequencies both lower and higher than fundamental themodel would not produce the frequency-dependent core behaviour. However, for these casestudies, there was no need to include frequency-dependent effects.For future work, there is a need to further test the core models against other fieldmeasurements, to fully determine the importance of frequency-dependent core models forpractical applications. There is also a possibility of computing saturation curves of deltaconnected transformers directly without iterations. This would save computer time, andmake the solution algorithm more reliable.References1. P.A.Abetti, Bibliography on the Surge Performance of Power Transformers andRotatingMachines, Trans.ATEE, vol. 77, 1958, PP. 1150-1168.2. C. Degeneff, A Methodfor Constructing Terminal Modelsfor Single-Phase N-WindingTransformers, Paper A78 539-9, presented at IEEE PES Summer Meeting, Los Angeles,Calif., July 16-21, 1978.3. L. Nakra, Behaviour of Circuits with Ferromagnetic Nonlinearity, Chapter 1, Ph. D.Thesis, McGill University, Montreal, August 1973.4. H. W. Dommel, Electromagnetic Transients Program Reference Manual, Section 6,Department of Electrical Engineering-The University of British Columbia, Vancouver,1986.5. F. L. Alvarado, Eletromagnetic Transients Program (EIVITP), EPRI, EL-4561, Vol. 3,The University ofWisconsin at Madison, Madison, Wisconsin, June 1989.6. W. F. Tinney, Compensation Methodsfor Network Solutions by Optimally Ordered Triangular Factorization, IEEE Trans. Power App. Syst., vol. PAS-91, Jan.fFeb. 1972, pp.123-127.7. J.G. Frame, N. Mohan and T. Liu, Hysteresis Modeling in an Electromagnetic Transients Program, IEEE, Trans. Power App. Syst.,vol. PAS-lOl, Sept. 1982, pp. 3403-3412.8. B. Mork and K. S. Rao, Modeling Ferroresonance with EMTP, EMTP-Newsletter,vol. 3, No. 4, May 1983, pp. 2-7.9. E. P. Dick and W. Watson, Transformer Modelsfor Transient Studies Based on FieldMeasurements, IEEE Trans. Power App. Syst, vol. PAS-100, No. 1, Jan. 1981, pp. 409-419.10.N. Germay, S. Mastero and J. Vroman, Review ofFerroresonance Phenomena in HighVoltage Power System and Presentation of a Voltage Transformer Model forPredetermining Them, CIGRE’, Paris, paper 33-18, 197410410511 .G. Biorci and D. Pescetti, Analytic Theory of the Behaviour ofFerromagnetic Materials, Nuovo Cimento, vol. 7, No. 6, 1958, PP. 829-842.12. I. D. Mayergoyz, MathematicalModels ofHysteresis, Springer-Verlag, 199113 .J.L. Binard, Hysteresis Model For Power Transformer Transient Simulation Program,EMTP Newsletter, Vol. 7, No. 3, Sept. 1987.14.J.G. Santesmases, J. Ayala, S. H. Cachero, Analytical Approximation of DynamicHysteresis Loops and its Application to a Series Ferroresonant Circuit, Proc. TEE 117,No. 1, January 1970, pp. 234-240.15.L. 0. Chua and K. A. Stromsmoe, Lumped Circuit Models for Nonlinear InductorsExhibiting Hysteresis Loops, IEEE Trans. on Circuit Theory, vol. CT-17, No. 4, Nov.1970, PP. 564-574.16.E.J. Tarasievicz, A. S. Morched, Detailed Transformer Modelfor EMTP; Part 1: EddyCurrent Representation, EMTP Review, Vol. 3, No. 4, October 1989.17.E.J. Tarasieviez, A. S. Morched, A. Narang and E. P. Dick, Frequency Dependent EddyCurrentModelsfor Nonlinear Iron Cores, IEEE Transactions on Power Systems, Vol. 8,No. 2, May 1993,PP.588-97.18. J. Avila-Rosales and F. L. Alvarado, Nonlinear Frequency Dependent Modelfor Electromagnetic Transient Studies in Power Systems, IEEE Trans. Power App. Syst, vol.PAS-lOl, Nov. 1982, pp. 428 1-4288.19.J. Avila-Rosales and A. Semlyen, Iron Core Modelingfor Electrical Transients, IEEETrans. Power App. Syst., vol. PAS-104, No. 11, Nov. 1985, pp. 3 189-3194.20.F. De Leon and A. Semlyen, Time Domain Modeling of Eddy current Effects forTransformer Transients, IEEE Transactions on Power Delivery, Vol. 8, No. 1, January1993, pp.271-80.21.F. Keffer, The Magnetic Properties ofMaterials, Scientific American, vol.217, No. 3,pp. 222-234.22.G. E. Fish, Soft Magnetic Materials, Proceedings of the IEEE, Vol. 78, No. 6, June1990, pp. 947-972.23.1.. J. Nahin, Oliver Heaviside: Sage in Solitude, IEEE Press, New York, 1987, pp. 109.10624. M. Latour, Note on Losses In Sheet Iron at Radio Frequencies, Proceedings of theInstitute ofRadio Engineers, V.7, 1918, PP. 61-7125.T. H. O’Dell, Ferromagnetodynamics, Chapters 1 and 5, Memillan Press Ltd, London,1981. /26.H. J. Williams, W. Shockley and C. Kittel, Studies of Propagation Velocity of a Ferromagnetic Domain Boundary, Phys. Rev. 80, 1950, pp. 1090-109427.P. D. Agarwal and L. Rabins, Rigorous Solution ofEddy Current Losses in RectangularBarfor Single Plane Domain WallModel, Journal of Applied Physics, Suplement to vol.31 No. 5, May 1960, pp. 246S-248S.28.C. W. Chen, Magnetism and Metalurgy of Soft Magnetic Materials, North HolandPublishing Company, New York, 1977.29.F. Brailsford, Investigation of the Eddy Current Anomaly in Electrical Sheet Steels, 3.lEE, 1948, No. 95, Pt. 2, pp. 12 1-125.3 O.K. J. Overshott, I. Preece and J. E. Thompson, Magnetic Properties of Grain-OrientedSilicon Iron, Part 2 - Basic Experiments on the Nature of the Anomalous Loss Using anIndividual Grain, Proc. lEE, Vol. 115, No. 12, December 1968, pp.1840-1845.31 .T. Spooner, Properties and Testing of MagneticMaterials, McGraw-Hill, 1927.32.C. Heck, Magnetic Materials and Their Applications, (English translation from theGerman),Chapter 10, Butterworth & Co. Ltd, 1974, pp. 337.33.S.L. Burgwin, Measurement ofCore Loss andA.C. Permeability with the 25 cm EpsteinFrame. Proceedings, Am, Soc. Testing Mats., ASTEA Vol. 41, 1941 pp. 779-796.34.G. Herzer and H. Hilzinger, Recent Developments in Soft Magnetic Materials, PhysicaScripta, Vol. T24, 1988, pp. 22-28.35.A. J. Moses, Recent Advances In Experimental Methods For The Investigation ofSilicon Iron, Physica Scripta, Vol. T24, 1988, pp. 49-53.36.F. J. Wilkins and A. E. Drake, Measurement and Interpretation of Power Losses inElectrical Sheet Steel, Proc. IRE Vol 112, No. 4, April, 1965, pp. 77 1-78537.A. Basak and A.A.A. Qader, Fundamental and Harmonic Flux Behaviour In a JOOkVADistribution Tranfor,ner Core, IEEE Trans. on Mag., Vol. Mag -19, No.5, September1983, pp. 2100-2102.10738. Stanley H. Charap, Magnetic Hysteresis Model, IEEE Trans. on Mag., vol. Mag -10,No. 4, December 1974, PP. 1091-1096.3 9.T. Nakata, Numerical Analysis of Flux and Loss Distribution inElectrical Machinery,(Invited Paper), IEEE Trans. on Magnetics, Vol. Mag - 20, No. 5, September, 1984.40.R. J. Larry and L. R. Turner, Survey of Eddy Current Programs, IEEE Trans. onMagnetics, Vol. Mag -19, No. 6, November 1983,pp.2474-7.41.G.W. Swift, Power Transformer Core Behavior Under Transient Conditions, IEEETrans. Power App. Syst,vol. PAS-90, No.5, September/October, 1971, pp. 2206-2210.42.H. W. Dommel, A. Yan, R. J. Marcano and A. B. Milliani, Case Studies For Electromagnetic Transients, UBC Department of Electrical Engineering Internal Report,Vancouver, 1983.43 .Microtran Power System Analysis Corporation, Transient Analysis Program ReferenceManual, Vancouver, Canada, 1991.44.Westinghouse Electric Coorporation, Electric Utility Engineering Reference Book, Vol.3, Distribution Systems, Chapter 6, Pittsburgh, Pensylvania, 1959.45.S. Prusty and M. V.S. Rao, A Direct Piecewise Linearized Approach to Convert rmsSaturation Characteristic to Instantaneous Saturation Curve, IEEE Transactions onMagnetics, Vol. Mag-16, No.1, January 1980, pp. 156-160.46.W. L. A. Neves and H. W. Dommel, On Modelling Iron Core Nonlinearities, IEEETransactions on Power Systems, Vol. 8, No. 2, May 1993, pp. 4 17-425.47.W. L. A. Neves, H. W. Dommel, Saturation Curves ofDelta-Connected TransformersFrom Measurements, to appear in IEEE Transactions on Power Delivery. Paper 94 SM459-8 PWRD presented at IEEE PES Summer Meeting, July 24-28, 1994, SanFrancisco, CA.48.W. L. A. Neves, H. W. Dommel and Wilsun Xu, Practical Distribution TransformerModelsfor Harmonic Studies, to appear in IEEE Transactions on Power Delivery. Paper 94SM 406-9 PWRD presented at IEEE PES Summer Meeting, July 24-28, 1994, SanFrancisco, CA.49.A. Otter and W. Neves, Loss Measurement Programs at TRIUMF, Proceedings of TheKaon PDS Magnet Design Workshop, TRIUMF, Vancouver, October 3-5, 1988, Pp.132-135.10850.M. B. Stout, Basic Electrical Measurements, Section 16-8, New York, Prentice HallInc., 1950.5 1.S.N. Talukdar, J.K. Dickson, R.C. Dugan, M.J. Sprinzen, C.J: Lenda, On ModelingTransformer and Reactor Saturation Characteristics for Digital and Analog Studies,IEEE Trans. on PAS, vol. PAS-94, 1975, pp. 612-621.52.C. G. A. Koreman, Determination of the Magnetizing Characteristic of Three-PhaseTransformers in Field Tests, IEEE Transactions on Power Delivery, Vol. 4, No. 3, July1989, pp. 1779-1785.53. Leuven EMTP Center (LEC), ATP Rule Book, Section XIX-G, Revision July, 1987.54. CIGRE Working Group 36.05, Harmonic, Characteristic Parameters, Methods ofStudy, Estimates ofExisting Values in the Network, Electra No. 77, pp. 3 5-54.55.C57 Collection, Distribution, Power andRegulating Transformers, IEEE Press, 199156.Canadian Electrical Association Project RP-070-D-165, Power Line Carrier onDistribution Systems, Ontario Hydro’s Research.57.Wilsun Xu, Multiphase Harmonic Load Flow Program (MHLF) Reference Manual.Powertech Labs, Inc., February, 199158.Allan Greenwood, Electrical Transients in Power Systems, John Wiley & Sons Inc., NewYork, 1991.59.ANSIIIEEE C37.011-1979, IEEEApplication Guidefor Transient Recovery Voltage forAC-High Voltage Circuit Breakers Rated on a Symmetrical Current Basis.60.CIGRE Working Group 02 of Study Committee 13, Interruption of Small InductiveCurrents: Chapter 5- Part I, Electra No. 133, pp. 79-96.61.Ernst A. Guillemin,, Synthesis ofPassive Networks, Chapter 8, John Wiley & Sons Inc.,New York, 1957.62.Athanasios Papoulis, The Fourier Integral and Its Applications, McGraw Hill BookCompany Inc.,1962, pp. 195-196.63 .Jiri Vlacb, ComputerizedApproximation and Synthesis ofLinear Networks, Chapter IV,John Wiley & Sons Inc., New York, 1969.64.B. D. Cullity, Introduction to Magnetic Materials, Addison Wesley Publishing Company,1972.10965.J. J. Becker, Magnetization Changes and Losses in Conducting FerromagneticMaterials, Journal of Applied Physics, Vol. 34, No. 04 (Part 2), April, 1963, pp. 1327-1332.66.David F. Tuttle, Electric Networks Analysis ant Synthesis, Chapter Xl, McGraw Hill Inc.1965.67.United States Steel, Electrical Steel Sheets Engineering Manual, 4th Edition, 525William Penn Place, Pittsburgh 30, PA.68.William H. Press, Saul A. Telkoisky, William T. Vetterling, Brian P. Flannery, NumericalRecipes, The Art of Scient/Ic Computing, 2 Edition, Section 5.13, CambridgeUniversity Press, New York,1992.69.L. Stuehm, B. A. Mork and D. Mairs, Five Legged Core Transformer EquivalentCircuit, IEEE Transactions on Power Delivery, Vol. PWRD-4, July 1989, pp. 1786-1793.70. A. C. Franklin and D. P. Franklin, The J & P Transformer Book: A Practical Technologyof the Power Transformer, 11th Edition, Butterworths, 1983.71.A. E. Emanuel, Nonsinusoidal Waveforms in Power Systems .Basic Definitions, IEEETutorial Course 90EH0327-7-PWR.AppendixA /Orthogonality Between ‘r and IConsider the circuit of Figure 3.1 a. The voltage across the transformer terminals andits correspondent flux linkage can be written in the formv(O)=VksinO, (A.1)andA(O)=—AkcosO, (A.2)respectively.Let us use Fourier analysis to represent the current i(O) through the nonlinearresistance and the current i,(O) through the nonlinear inductance. Due to the odd symmetryof the v—i,. and A—i1 curves, Ir(O) and i,() will have only odd harmonic components in theformjr(0)thi sinO-t-a33O+••• +asinpO+•. (A.3)andi,(9)=b1cosO+339+ +bcospO+”., (A.4)w&te p is odd.110Appendix A. Orthogonality Between Ir and 11111The total current 4(9) is then:i,(9) 1r(O)+Ii(O), (A.5)i.e.,where y = arctan(b I as).Evaluating the rrns values of 4(0), i,(O and i (O, we haveIr, = ja + a + + +• •, (A. 7)(A.8)and_______________________________+a+b+••-, (A.9)respectively.From equations (A.7), (A. 8) and (A.9), it can be seen thati=r.+I. (A. 10)Appendix BMeasurement ofthe InitialMagnetization CurveThe initial magnetization curve is a plot of the locus of the DC symmetricalhysteresis loop tips for different peak values of magnetization. Figure B. 1 is the circuit usedto measure it.V> To AnalyzerFigure B. 1: Measurement of the initial magnetization curve.The magnetizing winding of the Epstein frame (primary winding) is connected to aDC power supply through a reversing switch S, ammeter and a decade resistance box R.The secondary winding is connected to a digital waveform analyzer where the voltageREpstein Frame112Appendix B. Measurement of the Initial Magnetization Curve 113waveform is obtained and numerically integrated to give the flux linkage across thesecondary winding.The Epstein frame is demagnetized before any measurement is taken. This isaccomplished by driving the core into saturation using alternating current at powerfrequency and gradually reducing the core excitation to zero.After demagnetization, R is set to provide a low current, and S is reversed severaltimes to assure the sample is in a definite hysteresis cycle (AA’ and A’A trajectories ofFigure B.2). Then, the first reading takes place. The voltage across the secondary winding ofthe Epstein frame is integrated and the flux difference between AA’ is obtained. This valueis divided by two and segment OA is plotted.C,Figure B.2: Hysteresis ioop locus.After the first reading, R is changed to give a slightly greater value of the current inthe primary winding and the process is repeated up to the desired limit.IAppendix CComputation of Triplen Harmonic ComponentsAlthough only the computation of triplen harmonics is needed, it is appropriate toshow the derivation of all odd harmonic components of the current that may be produced bysaturation curves during standard tests. The equations below are developed for nonlinearinductances. For nonlinear resistances, one just needs to replace A. by and L by R in allequations.Consider the piecewise nonlinear inductance of Figure 4.2c. For a sinusoidal fluxA(O) = Ak sin( 0), the current can be written in a Fourier series form containing oddharmonics only’. So,1(0)=1sin0+b3sin30-I-..+bsinp0+•., (C.1)where p is odd, andb =-fi(O)sinp0d0. (C.2)For segment k =2,b =±[J A20+ ‘‘2 sin —A1 }inocio].114Appendix C. Computation of Triplen Harmonic Components 1151 1li 1Substituting F =—=— and F2 into the equation above, the fundamentalL1 “current (p = ) and its harmonics (p> ) are:b1=F2A+-[(F—F2)(s(012+)cos]andb =--[(Fj—F2)(g(p,01A+-cospJ.Withs(0i)=.{0i _sin20i)andsin[(p— 1)0] sin[(p + 1)8]g(p, 0)= —1) — 2(p +1)Finally, for any segment k, one can write=+— r)(s01,)k + A., cosand=(r1— g(p,011)2+-1cosP0ii)wheres(0i)={0i ..Jsin2Oi)andsin[(p — 1)0k] sin[(p + 1)0k]g(p, 0,)= —1) — 2(p +1)AppendixDRationalApproximation ofthe Real Part of Y(s)D.1 Fitting ProcedureSimilar to equation (5.8), the frequency-dependent resistance could be expressedas a rational function in -s2R— (_s2+a?)(_s2+a). (-s+a)(s) - R(_s2 +b12)(—s + b). (s2 + b)’(D. 1)and with s=jw, R(co) is written in the form:R — K(c1c02+1)(c2w+1).. (co2+ 1)(a)— C (dm2+1)(d2w+1).. .(dw2+ 1) (D.2)Where K, C. = 1/a and d1 = 1/12 are positive real numbers, with i = 1,2, , n The right handside of (D. 1)is fitted to the known function R(o). The goal here is to compute K, c. andd1. The number of terms n will depend on the frequency range of interest. R(co) is fittedinterval by interval as shown in Figure D. 1 starting with the interval betweeno and Oi andending with the interval between o and co.116Appendix D. Rational Approximation of the Real Part ofY(s) 117R(co)The algorithm proceeds as follows:1 - For interval 1, write R(w) asR -K(cIO.2+1)Kd0KC(w)_ c(d+1) (do,2 +1)orKdC02 -dR(co)w2+K0 = R(w) (D.3)2 - Estimate . Take m frequency sample points between o1 and w . Build theoverdetermined system of equations (D.3) with only the parameters K0, Kd and d1unknown. A weighted linear least square fitting routine[68] is used to find these parameters(c = Kd /K0 is also computed). If the maximum error in the fitting is less than a predifinedvalue, proceed to next step, If not, oi is reduced by small steps & i and the system ofequations (D.3) is reevaluated.3-For interval 2 in Figure D.1, write R(a) as(c 2 + 1)(c Q)2 + 1)—(01IFigure D. 1: Frequency dependent resistance.max COAppendix D. Rational Approximation of the Real Part ofY(s) 118with K, c1 and d. already known from the previous step. Estimate w2 and find c2 and d2following the procedure of step 2, with the ni frequency samples now taken between oando.For the remaining intervals in Figure D. 1, c. and d1 are found in the same fashion and theconstants 1?,,, a and b of (D. 1) are determined.Experience has shown that a first estimate of w, 5o (with Ao),=0. 1ci) andsubsequent estimates of co+, 10 o (with Aw1 = 0.1wi for i 2) are usually very close tothe final values for an error of 3% and 100 frequency samples used as input parameters tothe fitting routine.D.2 G(o)) Obtainedfrom Lamination DataSteel manufactures may supply core loss vs. frequency curves for the mostcommom lamination grades subjected to constant flux amplitudes’. These data are generallyobtained on Epstein samples using a sine wave voltage source following the test procedureofASTM Standard Method A-34. With this information one can obtain either the frequency-dependent resistance R(co) or conductance G(w) as described next.In the sinusoidal steady state the average powerP1 is:P,=P(co). (D.4)1 To avoid confusion of trade names, the American Iron and Steel Institute has assigned AISI type numbersto electrical steel. These consist of the letter “M” (for magnetic material) followed by a number which, whenthe designations were originally made, was about to ten times the core loss in watt/lb at 15 kilogauss (1.5 T)and 60 Hz for 29 gage sheet (0.0 140 in). Core losses have since been reduced but the type numbers remain[59].Appendix D. Rational Approximation of the Real Part ofY(s) 119R(o) is obtained by:V2 AcoR(w)=2= i (D.5)where Vm and Am are the amplitude of the applied voltage and flux, respectively. Since theflux is kept constant, R(w) can be written in the form:R(w)=Kp(w). (D.6)It is convenient to normalize R(w) assuming that the resistance measured at the transformerrated frequency o is R(o)=1. So, the constant K in the equation above isP(o.)K=, (D.7)rand the normalized resistance RN(e) isP(O)r)2RN(ftI)=w2 P(w)(D.8)So, if the transformer lamination loss vs. frequency curve is known, an estimate of thefrequency-dependent core loss resistance can be made by scaling the normalized resistanceRN(w) to the resistance at rated frequency R(o). Then,R(co)=RN(co)• R(o.ir), (D.9)or for the conductance G(o)G(o.I)=R).Appendix D. Rational Approximation of the Real Part ofY(s) 120D.3 G(w) Obtainedfrom Standard TestsThe frequency-dependent core resistance could also be estimated from transformerstandard tests. The plot of loss/cycle can be crudelly approximated by a straight line, as /shown in Figure D.2.Loss/cyc:fr fFigure D.2 Transformer core loss curve at rated flux.Power transformer manufacturers usually supply two or three values of no-loadloss at different frequencies, with the flux kept at the rated level. From Figure D.2, the lossper cycle coud be written as1o.s j47 Jfh+wheref is the frequency in which the loss is measured, Jr is the rated frequency, Wh is thehysteresis loss per cycle and Weddy is eddy loss per cycle at the rated frequency.Taking (D. 11) and writing the loss for the angular frequency w2itf,Appendix D. Rational Approximation of the Real Part ofY(s) 1211(D.12)From (D.6) and (D. 12), the resistance R(o) is /2wR(o,)=2P =m(D.13)loss—[wh +w_Normalizing R(co), as done for (D.7) and (D.8),KoRN(w)=(1)’ D14ratio+—COrwhere ratio=Ww’Weddy andratio + 1The resistance and conductance are then:R(o)=RN(w)•R(cor), (D.15)and G(CO)=R(’).If the transformer manufacturer does not supply the core loss vs. frequency curve, a typicalratio Wl/Wed+ is commonly taken as unity [70]. In practice, the straight line of Figure D.2could be drawn using loss measurement data gathered from routine tests. For instance, itcould be obtained from the loss measurement taken at rated voltage and rated frequencyand, as recommended by ANSI C57. 12.90 (Induced Overvoltage Withstand Test), from theloss measurement taken at twice the rated frequency and double voltage amplitude.


Citation Scheme:


Citations by CSL (citeproc-js)

Usage Statistics



Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            async >
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:


Related Items